PARAMETRIC OPTIMIZATION OF FUSED DEPOSITION MODELING USING RESPONSE SURFACE METHODOLOGY

PARAMETRIC OPTIMIZATION OF FUSED DEPOSITION MODELING USING RESPONSE SURFACE METHODOLOGY
PARAMETRIC OPTIMIZATION OF FUSED DEPOSITION
MODELING USING RESPONSE SURFACE METHODOLOGY
A THESIS SUBMITTED IN PARTIAL FULFILLMENT
FOR THE REQUIREMENT FOR THE DEGREE OF
Master of Technology
in
Production Engineering
by
VEDANSH CHATURVEDI
Department of Mechanical Engineering
National Institute of Technology
Rourkela – 8
2009
PARAMETRIC OPTIMIZATION OF FUSED DEPOSITION
MODELING USING RESPONSE SURFACE METHODOLOGY
A THESIS SUBMITTED IN PARTIAL FULFILLMENT
FOR THE REQUIREMENT FOR THE DEGREE OF
Master of Technology
in
Production Engineering
by
VEDANSH CHATURVEDI
Under the guidance of
Dr. S. S. MAHAPATRA
Professor, Department of Mechanical Engineering
Department of Mechanical Engineering
National Institute of Technology
Rourkela – 8
2009
National Institute of Technology
Rourkela
CERTIFICATE
This is to certify that the thesis entitled, “PARAMETRIC OPTIMIZATION OF
FUSED MODELING USING RESPONSE SURFACE METHODOLOGY”
submitted by Vedansh Chaturvedi in partial fulfillment of the requirements for the
award of Master of Technology Degree in Mechanical Engineering with
specialization in Production Engineering at the National Institute of Technology,
Rourkela (deemed University) is an authentic work carried out by him under my
supervision and guidance.
To the best of my knowledge, the matter embodied in the thesis has not submitted
to any other University/Institute for the award of any degree or diploma.
Dr. S. S. Mahapatra
Date:
Dept. of Mechanical Engineering
National Institute of Technology
Rourkela – 769008
ACKNOWLEDGEMENT
I would like to express my deep sense of respect and gratitude toward my supervisor Dr. S. S.
Mahapatra, who not only guided the academic project work but also stood as a teacher and
philosopher in realizing the imagination in pragmatic way, I want to thank him for introducing
me for the field of Optimization and giving the opportunity to work under him. His presence and
optimism have provided an invaluable influence on my career and outlook for the future. I
consider it my good fortune to have got an opportunity to work with such a wonderful person.
I express my gratitude to Dr. R. K. Sahoo, Professor and Head, Department of Mechanical
Engineering, faculty member and staff of Department of Mechanical Engineering for extending
all possible help in carrying out the dissertation work directly or indirectly. They have been great
source of inspiration to me and I thank them from bottom of my heart. I like to express my
gratitude to Dr. Saurav Datta, Lecturer, Department of Mechanical Engineering , for his
valuable advice in carrying out Literature review.
I am especially indebted to my parents for their love, sacrifices and support. They are my
teachers after I came to this world and have set great example for me about how to live, study
and work.
VEDANSH CHATURVEDI
CONTENTS
TITLE
PAGE NO.
Abstract
iv
List of Figures
v
List of Tables
vii
Nomenclature
viii
1. An introduction of rapid prototyping process
1
1.1 Overview of rapid prototyping process
1
1.2 The basic process
2
1.3 Rapid prototyping technique
4
1.3.1 Stereolithography
4
1.3.2 Selective layer sintering
6
1.3.3 Laminated object manufacturing
8
1.3.4 Fused deposition modeling
11
1.4
Objective of Research Work
12
2. Literature review
15
3. Fused deposition modeling and ABS material
20
3.1 Fused deposition modeling
20
3.2 ABS material
23
3.3 Properties of ABS plastic
24
4. Response surface methodology
4.1
27
Response surface methodology and robust design
i
30
4.2
The sequential nature of the response surface methodology
31
4.3
Building empirical models
32
4.3.1 Linear regression model
32
4.3.2 Estimation of the parameter in linear regression model
33
4.3.3 Model adequacy checking
34
4.3.4 Properties of the least square estimation regression model
34
4.3.5 Residual analysis
36
Variable selection and model building in regression
37
4.4.1 Procedure for variable selection
38
4.4.2 All possible regression
38
4.4.3 Stepwise regression analysis
39
4.4
5. Specimen preparation, Experiment and analysis
41
5.1
Specimen preparation
41
5.2
Testing of specimen
43
5.3
Analysis of experiments
47
5.4
5.3.1
Analysis of experiment for tensile test
47
5.3.2
RSA for tensile test
49
5.3.3
Analysis of experiment for flexural test
53
5.3.4
RSA for flexural test
54
5.3.5
Analysis of experiment for impact test
58
5.3.6
RSA for impact test
59
Optimization of process parameter
ii
62
6. Grey- based taguchi method
64
6.1 Introduction of grey – based taguchi method
64
6.2 Grey relational analysis method
65
6.3 Response optimization of GRG and optimal parameter setting
75
7. Result, Conclusion and future scope
77
7.1 Result, Discussion
77
7.2 Conclusion
80
7.3
80
Future scope
Bibliography
81
iii
ABSTRACT
Fused deposition modeling (FDM) is a process for developing rapid prototype (RP) objects by
depositing fused layers of material according to numerically defined cross sectional geometry.
The quality of FDM produced parts is significantly affected by various parameters used in the
process. This dissertation work aims to study the effect of five process parameters such as layer
thickness, sample orientation, raster angle, raster width, and air gap on mechanical property of
FDM processed parts. In order to reduce experimental runs, response surface methodology
(RSM) based on central composite design is adopted. Specimens are prepared for tensile,
flexural, and impact test as per ASTM standards. Empirical relations among responses and
process parameters are determined and their validity is proved using analysis of variance
(ANOVA) and the normal probability plot of residuals. Response surface plots are analyzed to
establish main factor effects and their interaction on responses. Optimal factor settings for
maximization of each response have been determined. Major reason for weak strength of FDM
processed parts may be attributed to distortion within the layer or between the layers while
building the parts due to temperature gradient. Since RP parts are subjected to different loading
conditions, practical implication suggests that more than one response must be optimized
simultaneously. To this end, mechanical properties like tensile strength, bending strength, and
impact strength of the produced component are considered as multiple responses and
simultaneous optimization has been carried out with the help of response optimizer. Grey
relation has been employed to convert multiple responses into a single response for optimization
purpose. It is interesting to note that factor level settings for simultaneous optimization of all
responses significantly differ from optimization with single response.
iv
LIST OF FIGURES
FIGURE NO.
FIGURE TITLE
PAGE NO.
1.1
Stereolithography
5
1.2
Selective laser sintering
7
1.3
Laminated object manufacturing
10
1.4
Fused deposition modeling
12
3.1
FDM Vantage Machine
21
3.2
Head assembly of FDM Vantage SE
21
3.3
Showing the process parameter of FDM
23
5.1
Line diagram of specimen for tensile test
42
5.2
Line diagram of specimen for flexural test
42
5.3
Line diagram of specimen for impact test
42
5.4
Instron test of tensile specimen
44
5.5
Instron test of 3-point bending specimen
44
5.6
Charpy test of impact specimen
45
5.7
Response surface plots for tensile test
50
5.8
SEM Images of tensile failure of specimen
52
5.9
Response surface plots for flexural test
56
5.10
SEM Images of crack surface of flexural specimen
57
v
5.11
Response surface plots for impact test
61
5.12
SEM Images of broke impact test specimen
61
6.1
Sensitivity analysis for different distinguishing coefficients
72
6.2
Grey relational grade variation with number of experiments
72
vi
LIST OF TABLES
TABLE NO.
TABLE TITLE
PAGE NO.
3.1
ABS Material data sheet
25
5.1
Domain of experiments (factors and their level)
41
5.2
Experimental data obtained from the CCD runs
46
5.3
Estimated regression coefficients for tensile test
47
5.4
Analysis of variance for tensile test
48
5.5
Estimated regression coefficients for flexural test
53
5.6
Analysis of variance for flexural test
54
5.7
Estimated regression coefficients for impact test
58
5.8
Analysis of variance for impact test
59
5.9
Optimum factor levels and predicted response for individual strength 63
6.1
Normalization of the data (larger the better)
67
6.2
The deviation sequence
68
6.3
Calculation of grey relational coefficients
69
6.4
Grey relational grade
70
6.5
Response surface analysis for grey relational grade
73
6.6
Estimated regression coefficients for grey relational grade
74
6.7
Analysis of variance for grey relational grade
75
6.8
Comparison of parameter setting in individual and
76
Simultaneous optimization
vii
NOMENCLATURE
RP
Rapid prototyping
STL
Stereolithography
FDM
Fused deposition modeling
ABS
Acrylonitrile butadiene styrene
RSM
Response surface methodology
CCD
Central composite design
ANOVA
Analysis of variance
SS
Sum of square
MS
Mean sum of square
DF
Degree of freedom
SEM
Scanning electron microscope
GRA
Grey relational analysis
GRG
Grey relational grade
viii
CHAPTER 1
______________________________________________________________________________
AN INTRODUCTION TO
RAPID PROTOTYPING
1. Introduction
1.1 Overview of Rapid Prototyping:
The term rapid prototyping (RP) refers to a class of technologies that can automatically construct
physical models from Computer-Aided Design (CAD) data. These "three dimensional printers"
allow designers to quickly create tangible prototypes of their designs, rather than just twodimensional pictures. Such models have numerous uses. They make excellent visual aids for
communicating ideas with co-workers or customers. In addition, prototypes can be used for
design testing. For example, an aerospace engineer might mount a model airfoil in a wind tunnel
to measure lift and drag forces. Designers have always utilized prototypes; RP allows them to be
made faster and less expensively.
In addition to prototypes, RP techniques can also be used to make tooling (referred to as rapid
tooling) and even production-quality parts (rapid manufacturing). For small production runs and
complicated objects, rapid prototyping is often the best manufacturing process available. Of
course, "rapid" is a relative term. Most prototypes require from three to seventy-two hours to
build, depending on the size and complexity of the object. This may seem slow, but it is much
faster than the weeks or months required to make a prototype by traditional means such as
machining. These dramatic time savings allow manufacturers to bring products to market faster
and more cheaply. In 1994, Pratt & Whitney achieved "an order of magnitude [cost] reduction
[and] . . . time savings of 70 to 90 percent" by incorporating rapid prototyping into their
investment casting process.
At least six different rapid prototyping techniques are commercially available, each with unique
strengths. Because RP technologies are being increasingly used in non-prototyping applications,
1
the techniques are often collectively referred to as solid free-form fabrication; computer
automated manufacturing, or layered manufacturing. The latter term is particularly descriptive of
the manufacturing process used by all commercial techniques. A software package "slices" the
CAD model into a number of thin (~0.1 mm) layers, which are then built up one atop another.
Rapid prototyping is an "additive" process, combining layers of paper, wax, or plastic to create a
solid object. In contrast, most machining processes (milling, drilling, grinding, etc.) are
"subtractive" processes that remove material from a solid block. RP’s additive nature allows it to
create objects with complicated internal features that cannot be manufactured by other means.
Of course, rapid prototyping is not perfect. Part volume is generally limited to 0.125 cubic
meters or less, depending on the RP machine. Metal prototypes are difficult to make, though this
should change in the near future. For metal parts, large production runs, or simple objects,
conventional manufacturing techniques are usually more economical. These limitations aside,
rapid prototyping is a remarkable technology that is revolutionizing the manufacturing process.
1.2 The Basic Process
Although several rapid prototyping techniques exist, all employ the same basic five-step process.
The steps are:
1. Create a CAD model of the design
2. Convert the CAD model to STL format
3. Slice the STL file into thin cross-sectional layers
4. Construct the model one layer atop another
5. Clean and finish the model
2
CAD Model Creation: First, the object to be built is modeled using a Computer-Aided Design
(CAD) software package. Solid modelers, such as Pro/ENGINEER, tend to represent 3-D objects
more accurately than wire-frame modelers such as AutoCAD, and will therefore yield better
results. The designer can use a pre-existing CAD file or may wish to create one expressly for
prototyping purposes. This process is identical for all of the RP build techniques.
Conversion to STL Format: The various CAD packages use a number of different algorithms
to represent solid objects. To establish consistency, the STL (stereolithography), the first RP
technique) format has been adopted as the standard of the rapid prototyping industry. The second
step, therefore, is to convert the CAD file into STL format. This format represents a threedimensional surface as an assembly of planar triangles, "like the facets of a cut jewel." The file
contains the coordinates of the vertices and the direction of the outward normal of each triangle.
Because STL files use planar elements, they cannot represent curved surfaces exactly. Increasing
the number of triangles improves the approximation, but at the cost of bigger file size. Large,
complicated files require more time to pre-process and build, so the designer must balance
accuracy with manageability to produce a useful STL file. Since the .stl format is universal, this
process is identical for all of the RP build techniques.
Slice the STL File: In the third step, a pre-processing program prepares the STL file to be built.
Several programs are available, and most allow the user to adjust the size, location and
orientation of the model. Build orientation is important for several reasons. First, properties of
rapid prototypes vary from one coordinate direction to another. For example, prototypes are
usually weaker and less accurate in the z (vertical) direction than in the x-y plane. In addition,
part orientation partially determines the amount of time required to build the model. Placing the
shortest dimension in the z direction reduces the number of layers, thereby shortening build time.
3
The pre-processing software slices the STL model into a number of layers from 0.01 mm to 0.7
mm thick, depending on the build technique. The program may also generate an auxiliary
structure to support the model during the build. Supports are useful for delicate features such as
overhangs, internal cavities, and thin-walled sections. Each PR machine manufacturer supplies
their own proprietary pre-processing software.
Layer by Layer Construction: The fourth step is the actual construction of the part. Using one
of several techniques (described in the next section) RP machines build one layer at a time from
polymers, paper, or powdered metal. Most machines are fairly autonomous, needing little human
intervention.
Clean and Finish: The final step is post-processing. This involves removing the prototype from
the machine and detaching any supports. Some photosensitive materials need to be fully cured
before use. Prototypes may also require minor cleaning and surface treatment. Sanding, sealing,
and/or painting the model will improve its appearance and durability.
1.3 Rapid Prototyping Techniques
Most commercially available rapid prototyping machines use one of six techniques. At present,
trade restrictions severely limit the import/export of rapid prototyping machines, so this guide
only covers systems available in the U.S.
1.3.1 Stereolithography:
Stereolithography is an additive fabrication process utilizing a vat of liquid UV-curable
photopolymer "resin" and a UV laser to build parts a layer at a time. On each layer, the laser
beam traces a part cross-section pattern on the surface of the liquid resin. Exposure to the UV
laser light cures, or, solidifies the pattern traced on the resin and adheres it to the layer below.
4
After a pattern has been traced, the SLA's elevator platform descends by a single layer thickness,
typically 0.05 mm to 0.15 mm (0.002" to 0.006"). Then, a resin-filled blade sweeps across the
part cross section, re-coating it with fresh material. On this new liquid surface the subsequent
layer pattern is traced, adhering to the previous layer. A complete 3-D part is formed by this
process. After building, parts are cleaned of excess resin by immersion in a chemical bath and
then cured in a UV oven.
Stereolithography requires the use of support structures to attach the part to the elevator platform
and to prevent certain geometry from not only deflecting due to gravity, but to also accurately
hold the 2-D cross sections in place such that they resist lateral pressure from the re-coater blade.
Supports are generated automatically during the preparation of 3-D CAD models for use on the
stereolithography machine, although they may be manipulated manually. Supports must be
removed from the finished product manually; this is not true for all rapid prototyping
technologies.
Figure 1.1 Stereolithography
Application Range
 Parts used for functional tests.
5
 Manufacturing of medical models.
 Form –fit functions for assembly tests.
Advantages
 Possibility of manufacturing parts which are impossible to be produced
conventionally in a single process.
 Can be fully atomized and no supervision is required.
 High Resolution.
Disadvantages
 Necessity to have a support structure.
 Require labor for post processing and cleaning.
1.3.2 Selective Laser Sintering:
Selective laser sintering is an additive rapid manufacturing technique that uses a high power
laser (for example, a carbon dioxide laser) to fuse small particles of plastic, metal, ceramic, or
glass powders into a mass representing a desired 3-dimensional object. The laser selectively
fuses powdered material by scanning cross-sections generated from a 3-D digital description of
the part (for example from a CAD file or scan data) on the surface of a powder bed. After each
cross-section is scanned, the powder bed is lowered by one layer thickness, a new layer of
material is applied on top, and the process is repeated until the part is completed.Compared to
other rapid manufacturing methods, SLS can produce parts from a relatively wide range of
commercially available powder materials, including polymers (nylon, also glass-filled or with
other fillers, and polystyrene), metals (steel, titanium, alloy mixtures, and composites) and green
sand. The physical process can be full melting, partial melting, or liquid-phase sintering. And,
6
depending on the material, up to 100% density can be achieved with material properties
comparable to those from conventional manufacturing methods. In many cases large numbers of
parts can be packed within the powder bed, allowing very high productivity.SLS is performed by
machines called SLS systems; the most widely known model of which is the Sinterstation SLS
system. SLS technology is in wide use around the world due to its ability to easily make very
complex geometries directly from digital CAD data. While it began as a way to build prototype
parts early in the design cycle, it is increasingly being used in limited-run manufacturing to
produce end-use parts. One less expected and rapidly growing application of SLS is its use in art.
SLS was developed and patented by Dr. Carl Deckard at the University of Texas at Austin in the
mid-1980s, under sponsorship of DARPA. A similar process was patented without being
commercialized by R.F. Housholder in 1979.Unlike some other Rapid Prototyping processes,
such as Stereolithography (SLA) and Fused Deposition Modeling (FDM), SLS does not require
support structures due to the fact that the part being constructed is surrounded by unsintered
powder at all times.
Figure 1.2 Selective laser sintering
7
Application Range
 Visual Representation models.
 Functional and tough prototypes.
 cast metal parts.
Advantages
 Flexibility of materials used.
 No need to create a structure to support the part.
 Parts do not require any post curing except when ceramic is used.
Disadvantages
 During solidification, additional powder may be hardened at the border line.
 The roughness is most visible when parts contain sloping (stepped) surfaces.
1.3.3 Laminated object manufacturing:
Profiles of object cross sections are cut from paper or other web material using a laser. The paper
is unwound from a feed roll onto the stack and first bonded to the previous layer using a heated
roller which melts a plastic coating on the bottom side of the paper. The profiles are then traced
by an optics system that is mounted to an X-Y stage. After cutting of the layer is complete,
excess paper is cut away to separate the layer from the web. Waste paper is wound on a take-up
roll. The method is self-supporting for overhangs and undercuts. Areas of cross sections which
are to be removed in the final object are heavily cross-hatched with the laser to facilitate
removal. It can be time consuming to remove extra material for some geometry, however. In
general, the finish, accuracy and stability of paper objects are not as good as for materials used
with other RP methods. However, material costs are very low, and objects have the look and feel
8
of wood and can be worked and finished in the same manner. This has fostered applications such
as patterns for sand castings. While there are limitations on materials, work has been done with
plastics, composites, ceramics and metals. Some of these materials are available on a limited
commercial basis. Variations on this method have been developed by many companies and
research groups. For example, Kira's Paper Lamination Technology (PLT) uses a knife to cut
each layer instead of a laser and applies adhesive to bond layers using the xerographic process.
Solido Ltd. of Israel (formerly Solidimension) also uses a knife, but instead bonds layers of
plastic film with a solvent. There are also variations which seek to increase speed and/or material
versatility by cutting the edges of thick layers diagonally to avoid stair stepping. The principal
US commercial provider of laser-based LOM systems, Helisys, ceased operation in 2000.
However the company's products are still sold and serviced by a successor organization, Cubic
Technologies.
The process is performed as follows:
1. Sheet is adhered to a substrate with a heated roller.
2. Laser traces desired dimensions of prototype.
3. Laser cross hatches non-part area to facilitate waste removal.
4. Platform with completed layer moves down out of the way.
5. Fresh sheet of material is rolled into position.
6. Platform moves up into position to receive next layer.
7. The process is repeated.
9
Figure 1.3 Laminated object manufacturing
Application Range
 Visual Representation models
 Large Bulky models as sand casting patterns
Advantages
 Variety of organic and inorganic materials such as paper, plastic, ceramic,
composite can be used
 Process is faster than other processes
 No internal stress and undesirable deformations
 LOM can deal with discontinuities, where objects are not closed completely
Disadvantages
 The stability of the object is bonded by the strength of the glued layers.
 Parts with thin walls in the z direction can not be made using LOM
 Hollow parts can not be built using LOM
10
1.3.4 Fused Deposition Modeling
Fused deposition modeling, which is often referred to by its initials FDM, is a type of additive
fabrication or (sometimes called rapid prototyping/rapid manufacturing (RP or RM)) technology
commonly used within engineering design[1]. The technology was developed by S. Scott Crump
in the late 1980s and was commercialized in 1990. The FDM technology is marketed
commercially by Stratasys, which also holds a trademark on the term.
Like most other additive fabrication processes (such as 3D printing and stereolithography) FDM
works on an "additive" principle by laying down material in layers. A plastic filament or metal
wire is unwound from a coil and supplies material to an extrusion nozzle which can turn on and
off the flow. The nozzle is heated to melt the material and can be moved in both horizontal and
vertical directions by a numerically controlled mechanism, directly controlled by a computeraided manufacturing (CAM) software package [2]. The model or part is produced by extruding
small beads of thermoplastic material to form layers as the material hardens immediately after
extrusion from the nozzle.
Several materials are available with different trade-offs between strength and temperature
properties. As well as acrylonitrile butadiene styrene (ABS) polymer, the FDM technology can
also be used with polycarbonates, polycaprolactone, polyphenylsulfones and waxes. A "watersoluble" material can be used for making temporary supports while manufacturing is in
progress[3]. Marketed under the name Waterworks by Stratasys, this soluble support material is
quickly dissolved with specialized mechanical agitation equipment utilizing a precisely heated
sodium hydroxide solution.
11
Figure 1. 4 Fused deposition modeling
Application Range
 Conceptual modeling
 Fit, form applications and models for further manufacturing procedures
 Investment casting and injection molding
Advantages
 Quick and cheap generation of models
 There is no worry of exposure to toxic chemicals, lasers or a liquid chemical bath.
Disadvantages
 Restricted accuracy due to the shape of material used, wire is 1.27 mm diameter.
1.4 Objective of Research Work
The competition in world market for manufactured product has intensified tremendously in
recent years. It has become important for new products to reach the market as early as possible.
As a result reduction of product development cycle time is a major concern in industries for
achieving competitive advantage. Now days the focus of industries has shifted from traditional
product development methodology to accelerated or rapid fabrication techniques. Some of the
12
latest developments within the automotive industry have shown how emerging rapid prototyping
and manufacturing (RP&M) technologies can be used to reduce lead time in the prototype
development process. The main benefit rapid prototyping (RP) technologies offer as compared to
conventional subtractive and formative manufacturing process is that virtually any complex
geometry can be built in a layer wise manner directly from CAD model of part without the need
for tooling using a nearly fully automated process. This ability to fabricate complex geometry at
no extra cost is virtually unheard in traditional manufacturing, where there is direct link between
cost of component and complexity of design. On the other hand absence of tooling means that
manufacturing inputs are not required in order to design parts and products . For example if RP is
used to manufactured the part which was conventionally manufactured by injection moulding,
considerations for draft angles, ejection pins and gates marks, wall thickness, sharp corners, weld
lines and parting lines is not important for part design. This directly means whatever can be
designed it can be manufactured. That is optimal design can be selected for manufacturing
without considering the feasibility of their production in terms of available manufacturing
technology. Incorporating features such as undercuts, blind holes, screws in process like
injection moulding often requires expensive tooling, extensive tool setups and testing runs and
inevitably leads to undesirable lead times and costs. Also there is a threshold limit for minimum
production level which has to be cross to offset the cost of tooling. This result in high volume
manufacturing to compensate the tooling cost. Whereas the possibility of producing highly
complex, cost effective custom parts is apparent in RP . Another noted advantage of RP is their
ability to produce functional assembly by consolidating sub assemblies into single unit at the
computer aided design (CAD) stage thus reducing the part count, handling time storage
requirement and without considering the mating and fit problem. RP allows the deposition of
13
multiple materials in any location or combination that the designer requires. This has potentially
enormous implications for the functionality and aesthetics that can be designed into parts .
Having such enormous advantages one of the biggest hindrance in the full scale application of
RP technologies is available materials and their properties, which substantially differ from the
properties of generally used materials. To overcome this limitation one approach is to develop
new materials which can be used by RP machines and have properties superior or as par with
conventional materials. Another procedure is to suitably adjust the process parameters for RP
part fabrication for maximum improvement in the properties. Number of researchers contributed
in this second approach. Their works reveal that properties of RP parts are function of various
process related parameters and can be significantly improved with their proper adjustment. Since
mechanical strength is an important requirement for the functional part there is great need to
improve them. With this aim in mind the present study focus on the mechanical properties viz.
tensile, flexural and impact strength of part fabricated using fused deposition modeling (FDM)
technology and derive the quantitative relation between the processing parameters and
mechanical strength so that the mechanical response of the processed part must be predictable
over the allowable range of parameter.
14
CHAPTER 2
______________________________________________________________________________
A BRIEF LITERATURE REVIEW
2. Literature review
Ahn et al. [4] Uses design of experiment method and concluded that the air gap and raster
orientation affect the tensile strength of FDM processes part where as raster width, model
temperature and colour have little effect. They further compare the measured tensile strength of
FDM part processed at different raster angles and air gap with the tensile strength of injection
moulded part. Material use for both type of fabrication is ABSP400. With zero air gap FDM
specimen tensile strength lies between 10%-73% of injection moulded part with maximum at 0°
and minimum at 90° raster orientation with respect to loading direction. But with negative air
gap there is significant increase in strength at respective raster orientation but still it is less than
the injection moulded part. All specimens failed in transverse direction except for specimen
whose alternate layer raster angle varies between 45° and -45°. This type of specimen failed
along the 45° line. Compression test on the specimen build at two different orientations revealed
that this strength is higher than the tensile strength and lies between 80 to 90% of those for
injection moulded part. Also specimen build with axis perpendicular to build table shows less
compressive strength as compared to specimen build with axis parallel to build table. Based on
these observations it was concluded that strength of FDM processed part is anisotropic.
Es Said et al. [5] Study the effect of raster angle on the tensile, bending and impact properties
of FDM ABSP400 part made using FDM1650 machine. There observations indicate that raster
orientation effect the strength as polymer molecules align themselves along the direction of flow.
Also FDM follows phase change for constructing solid model from solid filament extruded from
nozzle tip in semi molten state and solidify in a chamber maintain at particular temperature. As a
15
result volumetric shrinkage takes place which results in weak interlayer bonding and cause
porosity which reduce the load bearing area.
Lee et al. [6] Performed experiments on cylindrical parts made using three RP processes FDM,
3D printer and nano composite deposition (NCDS) to study the effect of build direction on the
compressive properties. Experimental results show that compressive strength is 11.6% higher for
axial FDM specimen as compared to transverse FDM specimen. In 3D printing, diagonal
specimen possesses maximum compressive strength in comparison to axial specimen. For
NCDS, axial specimen showed compressive strength 23.6% higher than that of transverse
specimen. Out of three RP technologies, parts built by NCDS are most affected by the build
direction.
Khan et al. [7] concluded that layer thickness, raster angle and air gap are found to be
significantly affect the elastic performance of the compliant FDM ABS prototype.
Wang et al. [8], in their work, has mentioned that as extruded material from nozzle cools from
its glass transition temperature to chamber temperature inner stresses will develop particularly
due to uneven deposition speed. These inner stresses will cause the inter layer and the intra layer
deformation which will result in cracking, de-lamination or even part fabrication failure. Thus
affect the part strength and size. They propose the mathematical model to study the effect of total
number of layers, stacking section length, and chamber temperature on the above mentioned
deformations. They concluded that as the total number of layers increase deformation will
decrease rapidly but decreasing tendency will become slow after certain number of layers, higher
stacking section lengths will produce large deformations and as chamber temperature will
increase deformation will decrease and become zero at the glass transition temperature of
16
material. Based on these results they propose that material use for part fabrication must have
lower glass transition temperature and linear shrinkage rate. Also the extruded fiber length must
be small.
Bellehumeur et al. [9] experimentally assessed the bond quality between adjacent filaments
and their failure under flexural loading. Experimental results showed that both the envelope
temperature and variations in the convective conditions with in the building chamber have strong
effect on the meso-structure and the overall quality of bond strength. On line measurements of
the cooling temperature profiles reveals that temperature profile of bottom layers rises above the
glass transition temperature followed by rapid decrease as the extrusion head moves away from
the position of placement of thermoset and minimum temperature increase with the number of
layers. Microphotograph of the cross sectional area shows diffusion of adjacent filaments is more
in lower layers as compared to upper layers for the face of specimen with higher number of
layers.
Chou and Zang [10], in their work, simulated the FDM process using finite element analysis
(FEA) and analyzes the effect of tool path patterns on residual stresses and part distortions. At
each layer stress starts to accumulate at the locations of initial deposition and at tool path turning
point. During the deposition process, the residual stress is smallest for most recently activated
elements as compare to earlier activated element. The residual effect on the bottom surface of
each layer corresponds to stress concentration pattern of its bottom layer. For the long raster
pattern stress concentration characteristic is aligned along the length side and along the width
side for the short raster deposition. Thus the maximum stress zone shifts from the center of the
part towards the length side and width side for the long and short raster pattern respectively.
Simulated results are found to be in agreement with experimental results of the distortion in part
17
except the magnitude of distortion is more than the expected and this may be due to simplified
material properties and boundary conditions assumed during simulation.
Above mention work reveals that the mechanical properties of FDM processed part exhibit
anisotropy and are sensitive to the processing parameters that affect the meso-structure and fibreto-fibre bond strength. Also un-even heating and cooling cycles due to inherent nature of FDM
build methodology results in stress accumulation in the build part and these stress concentration
regions will also affect the strength. It is also observed that all the researches in FDM strength
modeling is basically devoted to study the effect of processing conditions on the part strength
with no significant effort made to develop the strength model in terms of FDM process
parameters so as to predict in advance the strength of component for practical application.
Anitha et al. [11], in their result, revealed several interesting features of the FDM process
Only the layer thickness is effective to 49.37% at 95% level of significance. But on pooling, it
was found that the layer thickness is effective to 51.57% at 99% level of significance. The other
factors, road width and speed , contributes to 15.57 and 15.83% at 99% level of significance,
respectively. The significance of layer thickness is further strengthened by the correlation
analysis. Which indicates a strong inverse relationship with surface roughness.
According to the S/N analysis, the layer thickness is most effective when it is at level
3(0.3556mm), the road width at level 1(0.537mm) and the speed of deposition at level 3
(200mm).According to this trials , sample 18 was found to give the best results.
Agrawal et al. [12] In this work, the concept of stochastic modelling of tolerances and
clearances has been extended to RP processes. Using the unified approach for RP processes, the
mechanical error in the FD process has been studied. A methodology has been developed to
analyse the mechanical error at the nozzle tip of the FD process for input values of the tolerances
18
and clearances, where the links and hinges are produced on a mass scale. Closed-form
expressions have been derived to find the mechanical error in the coordinates of a point on the
work surface. It is observed that the influence coefficients of the z coordinate of a point on the
work surface have a larger magnitude than those of the x and y coordinates. The three-sigma
bands obtained in tracing a few example curves by the nozzle tip are plotted. The variances and
their sum are listed in a table to show their variation across the work surface.
The overall error is found to vary appreciably across the work surface. The error is minimum at
the front-left end of the work surface and maximum at the rear-right end. The methodology can
be extended for the optimal allocation of tolerances and clearances to reduce the cost of
manufacturing.
Pandey et al. [13] In this research they found Orientation for part deposition is one of the
important factors as it affects average part surface roughness and production time. In the present
work, two objective functions, namely average part surface roughness and build time, are
formulated.
NSGA-II is successfully used to determine a set of pareto optimal solutions for part deposition
orientation for the two contradicting objectives. It can be seen from the results obtained for
different parts that there exist two limiting situations. One is minimum average part surface
roughness with maximum production time and another is minimum production time with
maximum average part surface roughness. The developed system of part deposition orientation
determination also gives a set of intermediate solutions in which any solution can be used
depending upon the pre- ference of user for the two objectives. The present system can be used
for any class of component, which may be a freeform or a regular object.
19
CHAPTER 3
______________________________________________________________________________
FUSED DEPOSITION MODELLING
AND ABS MATERIAL
3. Fused deposition modeling and ABS material
3.1 Fused Deposition Modeling
FDM is one of the RP technology developed by Stratasys, USA (Figure 3.1). But unlike other RP
systems which involve an array of lasers, powders, resins, this process uses heated thermoplastic
filaments which are extruded from the tip of nozzle in a temperature controlled environment. For
this there is a material deposition subsystem known as head (Figure 3.2) which consist of two
liquefier tips. One tip for model material and other tip for support material deposition both of
which works alternatively. The article forming material is supplied to the head in the form of a
flexible strand of solid material from a supply source (reel). One pair of pulleys or rollers having
a nip in between are utilized as material advance mechanism to grip a flexible strand of modeling
material and advance it into a heated dispensing or liquefier head. The material is heated above
its solidification temperature by a heater on the dispensing head and extruded in a semi molten
state on a previously deposited material onto the build platform following the designed tool path.
The head is attached to the carriage that moves along the X-Y plane. The build platform moves
along the Z direction. The drive motion are provided to selectively move the build platform and
dispensing head relative to each other in a predetermined pattern through drive signals input to
the drive motors from CAD/CAM system. The fabricated part takes the form of a laminate
composite with vertically stacked layers, each of which consists of contiguous material fibres or
rasters with interstitial voids. Fibre-to-fibre bonding within and between layers occurs by a
thermally-driven diffusion bonding process during solidification of the semi-liquid extruded fibre
[14].
20
FDM Vantage uses insight software to import STL file automatically slice the file, generate
necessary support structure and material extrusion path [15].
Power required
- 230 V,AC
Motor
- 50/60 Hz,3Ф
Max. room temperature
Size of the system
- 29.3oC
- 1277mm wide X 874 mm deep X 1950 mm hight
Figure 3.1 FDM Vantage machine SE
Inlet passage
Roller
Support liquefier
Model liquefier
Support tip
Model tip
Figure 3.2 Head Assembly of FDM Vantage SE
21
Main process parameters involved in part fabrication are [3.3]:
1. Orientation: Part builds orientation or orientation referrers to the inclination of part in a
build platform with respect to X, Y, Z axis. Where X and Y-axis are considered parallel
to build platform and Z-axis is along the direction of part build.
2. Layer thickness: It is a thickness of layer deposited by nozzle and depends upon the type
of nozzle used.
3. Part Fill Style: Determines the manner in which nozzle will deposit the material in a
single layer of a part. There are two types of part fill methods:
Perimeter Raster: Outer boundary is contour (perimeter) and internal is filled with raster.
Contours to depth: Other then the outer contour additional contours are provided to fill
the inner region and remaining inner region is filled with raster. The number of additional
contours is determined by the depth of contours value. By default depth of contour is
twice the contour width to produce one contour.
4. Contour width: The width of contour deposited by nozzle.
5. Part raster width (raster width): Width of raster pattern used to fill interior regions of
part curves.
6. Part interior style: Determine how the interior area in each layer is filled. There are two
methods:
Solid normal: Fills the part completely
Sparse: Semi hollow interior (honeycomb structure), minimize the amount of material
used.
7. Visible surface: This feature improves the part external appearance.
8. Raster angle: It is a direction of raster relative to the x-axis of build table.
22
9. Shrinkage factor: Shrinkage factor applied in the x, y and z direction.
10. Perimeter to raster air gap: The gap between inner most contours and the edge of the
raster fill inside of the contour.
11. Raster to raster gap (air gap): It is the gap between two adjacent rasters on same layer.
Figure 3.3 showing process parameter of FDM
3.2 ABS (Acrylonitrile butadiene styrene) material:
Acrylonitrile Butadiene Styrene (ABS) chemical formula (C8H8· C4H6·C3H3N)n) is a common
thermoplastic used to make light, rigid, molded products such as piping (for example Plastic
Pressure Pipe Systems), musical instruments (most notably recorders and plastic clarinets), golf
club heads (used for its good shock absorbance), automotive body parts, wheel covers,
enclosures, protective head gear, buffer edging for furniture and joinery panels, airsoft BBs and
toys, including Lego bricks. ABS plastic ground down to an average diameter of less than 1
micrometer is used as the colorant in some tattoo inks. Tattoo inks that use ABS are extremely
vivid. This vividness is the most obvious indicator that the ink contains ABS, as tattoo inks
rarely list their ingredients [4] .
It is a copolymer made by polymerizing styrene and acrylonitrile in the presence of
polybutadiene. The proportions can vary from 15 to 35% acrylonitrile, 5 to 30% butadiene and
23
40 to 60% styrene. The result is a long chain of polybutadiene criss-crossed with shorter chains
of poly(styrene-co-acrylonitrile). The nitrile groups from neighboring chains, being polar, attract
each other and bind the chains together, making ABS stronger than pure polystyrene. The styrene
gives the plastic a shiny, impervious surface. The butadiene, a rubbery substance, provides
resilience even at low temperatures. ABS can be used between −25 and 60 °C. The properties are
created by rubber toughening, where fine particles of elastomer are distributed throughout the
rigid matrix.
Production of 1 kg of ABS requires the equivalent of about 2 kg of oil for raw materials and
energy. It can also be recycling.
3.3 Properties of ABS plastic
ABS is derived from acrylonitrile, butadiene, and styrene. Acrylonitrile is a synthetic monomer
produced from propylene and ammonia; butadiene is a petroleum hydrocarbon obtained from the
C4 fraction of steam cracking; styrene monomer is made by dehydrogenation of ethyl benzene a hydrocarbon obtained in the reaction of ethylene and benzene. The advantage of ABS is that
this material combines the strength and rigidity of the acrylonitrile and styrene polymers with the
toughness of the polybutadiene rubber. The most important mechanical properties of ABS are
resistance and toughness. A variety of modifications can be made to improve impact resistance,
toughness, and heat resistance. The impact resistance can be amplified by increasing the
proportions of polybutadiene in relation to styrene and also acrylonitrile although this causes
changes in other properties. Impact resistance does not fall off rapidly at lower temperatures.
Stability under load is excellent with limited loads [7].
Even though ABS plastics are used largely for mechanical purposes, they also have good
electrical properties that are fairly constant over a wide range of frequencies. These properties
24
are little affected by temperature and atmospheric humidity in the acceptable operating range of
temperatures. The final properties will be influenced to some extent by the conditions under
which the material is processed to the final product; for example, molding at a high temperature
improves the gloss and heat resistance of the product whereas the highest impact resistance and
strength are obtained by molding at low temperature.
ABS polymers are resistant to aqueous acids, alkalis, concentrated hydrochloric and phosphoric
acids, alcohols and animal, vegetable and mineral oils, but they are swollen by glacial acetic
acid, carbon tetrachloride and aromatic hydrocarbons and are attacked by concentrated sulfuric
and nitric acids. They are soluble in esters, ketones and ethylene dichloride.
The aging characteristics of the polymers are largely influenced by the polybutadiene content,
and it is normal to include antioxidants in the composition. On the other hand, while the cost of
producing ABS is roughly twice the cost of producing polystyrene, ABS is considered superior
for its hardness, gloss, toughness, and electrical insulation properties. However, it will be
degraded (dissolve) when exposed to acetone. ABS is flammable when it is exposed to high
temperatures, such as a wood fire. It will "boil", then burst spectacularly into intense, hot flames.
Table 3.1 ABS material data sheet
Properties
Specifications
Amorphous
1.05
0.27
20
29.64
62.05
63.43
2068.48
8.94
R110
Excellent
Structure
Specific density
Water absorption rate (%)
Elongation (%)
Tensile strength (MPa)
Compression strength (MPa)
Flexural strength (MPa)
Flexural modulus (MPa)
Impact (joules)
Hardness
Ultrasonic welding
25
Bonding
Machining
Min. utilization temperature (deg. C)
Max. utilization temperature (deg .C)
Melting point (deg.C)
Coefficient of expansion
Arc resistance
Dielectric strength (KV/mm)
Transparency
UV Resistance
Chemical resistance
Excellent
Good
-40
90
105
0.000053
80
16
Translucent
Poor
Good
26
CHAPTER 4
______________________________________________________________________________
RESPONSE SURFACE METHODOLOGY
4. Response surface methodology
Response surface methodology is very useful and modern technique for the prediction and
optimization of machining performances. In the present study, the strength of ABS material part
made by fused deposition modelling machine has been predicted and also process parameters
have been optimized by RSM. In this chapter, overview of RSM has been discussed. Response
surface methodology (RSM) is a collection of statistical and mathematical techniques useful for
developing, improving, and optimizing processes. The most extensive applications of RSM are
in the particular situations where several input variable potentially influence some performance
measure or quality characteristic of the process. This performance measure or quality
characteristic is called the response. The input variables are sometimes called independent
variables. The field of response surface methodology consists of the experimental strategy for
exploring the space of the process or independent variables, Empirical statistical modelling to
develop an approximated relationship between the yield and the process variables. Also, with the
help of response surface methodology, optimization can be done for finding the values of the
process variables that produce desirable values of the response [16].
In general, the relationship between the response y and independent variables ξ1, ξ2, …., ξk is,
Y= f(ξ1,ξ2,………………ξk) + ε
………….……………(4.1)
where ε includes effects such as measurement error on the response, background noise, the
effect of other variables, and so on. Usually ε is treated as a statistical error, often assuming it to
have a normal distribution with mean zero and variance σ2. Then,
E(y) = η =E[f(ξ1,ξ2,…………ξk)] + E(ε) =f(ξ1, ξ2,………..ξk)
27
……………...(4.2)
The variables ξ1, ξ2,…., ξk in equation (4.2) are usually called the natural variables, because they
are expressed in the natural units of measurement, such as degrees Celsius, pounds per square
inch, etc. In much RSM work, it is convenient to transform the natural variables to coded
variables x1, x2, …. ,xk, which are usually defined to be dimensionless with mean zero and the
same standard deviation. In terms of the coded variables, the response function equation (4.2)
can be written as,
η =f(x1,x2……….xk)
…………………………….(4.3)
Because the form of the true response function is unknown, it should be approximated. In fact,
successful use of RSM is critically dependent upon the experimenter’s ability to develop a
suitable approximation. Usually, a low-order polynomial in some relatively small region of the
independent variable space is appropriate. In many cases, either a first-order or a second-order
model is used. The first-order model is likely to be appropriate when the experimenter is
interested in approximating the true response surface over a relatively small region of the
independent variable space in a location where there is little curvature in response function. For
the case of two independent variables, the first-order model in terms of the coded variables is
given by,
η = β0 +β1x1 +β2x2
…………………………….……………..
(4.4)
The form of the first-order model in equation (4.4) is sometimes called a main effects model,
because it includes only the main effects of the two variables x1 and x2. If there is an interaction
between these variables, it can be added to the model easily as expressed below:
η = β0 +β1x1 +β2x2+β12x1x2
28
……………………….……… (4.5)
This is the first-order model with interaction. Adding the interaction term introduces curvature
into the response function. Often the curvature in the true response surface is strong enough that
the first-order model (even with the interaction term included) is inadequate. A second-order
model will likely be required in these situations. For the case of two variables, the second-order
model is:
η = β0 +β1x1 +β2x2+β11x21+β22x2 2+β12x1x2
……………………..…… (4.6)
This model would likely be useful as an approximation to the true response surface in a relatively
small region.
The second-order model is widely used in response surface methodology for several reasons:
the second-order model is very flexible. It can take on a wide variety of functional forms,
so it will often work well as an approximation to the true response surface.
It is easy to estimate the parameters in the second-order model. The method of least
Squares can be used for this purpose.
There is considered to be practical experience indicating that second-order models work
well in solving real response surface problems. In general, the first-order model is:
η = β0 +β1x1 +β2x2+…………+βkxk
…………………………….…………(4.7)
And the second-order model is
η = β0 +
+
………………….………………. (4.8)
+
Finally, it should be noted that there is a close connection between RSM and linear regression
analysis. For example, say, the following model is considered:
η = β0 +β1x1 +β2x2+…………+βkxk + ε
29
……………..………………(4.9)
The β’s are a set of unknown parameters. To estimate the values of these parameters, the
experimental data must be needed.
4.1. Response Surface Methodology and Robust Design:
RSM is an important branch of experimental design. It is also a critical technology in developing
new processes and optimizing their performance. The objectives of quality improvement,
including reduction of variability and improved process and product performance, can often be
accomplished directly using RSM.
It is well known that variation in key performance
characteristics can result in poor process and product quality. During the 1980s, considerable
attention was given to process quality, and methodology was developed for using experimental
design, specifically for the following:
For designing or developing products and processes so that they are robust to component
variation.
For minimizing variability in the output response of a product or a process around a
target value.
For designing products and processes so that they are robust to environment conditions.
By robust means that the product or process performs consistently on target and is
relatively insensitive to factors that are difficult to control.
Professor Genichi Taguchi used the term robust parameter design (RPD) to describe his
approach to this important problem. Essentially, robust parameter design prefers to reduce
process or product variation by choosing levels of controllable factors (or parameters) that make
the system insensitive (or robust) to changes in a set of uncontrollable factors. These
uncontrollable factors represent most of the sources of variability. Taguchi referred to these
uncontrollable factors as noise factors. In RSM, it is assumed that these noise factors are
30
uncontrollable in the field, but can be controlled during process development for purposes of a
designed experiment.
Considerable attention has been focused on the methodology advocated by Taguchi, and a
number of flaws in his approach have been discovered. However, the framework of response
surface methodology allows easily incorporate many useful concepts in his philosophy.
4.2 The Sequential Nature of the Response Surface Methodology:
Most applications of RSM are sequential in nature. They are briefly discussed below:
Phase 0: At first, some ideas should be generated concerning which factors or variables are
likely to be important in response surface study. It is usually known as a screening experiment.
The objective of factor screening is to reduce the list of candidate variables to a relatively few so
that subsequent experiments will be more efficient and require fewer runs or tests. The purpose
of this phase is the identification of the important independent variables.
Phase 1: The objective of the experiment is to determine if the current settings of the
independent variables result in a value of the response that is near the optimum. If the current
settings or levels of the independent variables are not consistent with optimum performance, then
a set of adjustments must be done to the process variables that will move the process toward the
optimum. This phase of RSM makes considerable use of the first-order model and an
optimization technique called the method of steepest ascent /descent.
Phase 2: When the process is near the optimum, it is required to develop a model that will
accurately approximate the true response function within a relatively small region around the
optimum. As the true response surface usually exhibits curvature near the optimum, a secondorder model (or perhaps some higher-order polynomial) should be used. Once an appropriate
approximated model has been obtained, this model may be analyzed to determine the optimum
31
conditions for the process. This sequential experimental process is usually performed within
some region of the independent variable space called the operability region or experimentation
region or region of interest.
4.3 Building Empirical Models:
4.3.1 Linear regression model:
In the practical application of RSM, it is necessary to develop an approximated model for the
true response surface. The true response surface is typically driven by some unknown physical
mechanism. The approximated model is based on observed data from the process or system and
it is an empirical model. Multiple regressions is a collection of statistical techniques useful for
building the types of empirical models required in RSM.
The first-order multiple linear regression models with two independent variables is:
Y = β0+β1x1+β2x2+ε
………………………….………………….(4.10)
The independent variables are often called predictor variables or regressors. The term “linear” is
used because equation (4.10) is a linear function of the unknown parameters β0, β1 and β2.
In general, the response variable y may be related to k regressor variables. The model is given
by:
Y = β0+β1x1+β2x2 +……..+ βkxk+ε
………………………………..(4.11)
This is called a multiple linear regression model with k regressor variables. The parameters
βj, j=0,l, ...k, are called the regression coefficients. Models those are more complex in
appearance than equation (4.11) may often be analyzed by multiple linear regression techniques.
For example, adding an interaction term to the first-order model in two variables, the model
becomes:
Y= βo+β1x1+β2x2+β12x1x2+ε
……………………..………………(4.12)
32
As another example, considering the second-order response surface model in two variables,
the model becomes:
η = β0 +β1x1 +β2x2+β11x21+β22x2 2+β12x1x2 + ε
……………………(4.13)
In general, any regression model that is linear in the parameters (the β-values) is a linear
regression model, regardless of the shape of the response surface that it generates.
4.3.2 Estimation of the parameters in linear regression models:
The method of least squares is typically used to estimate the regression coefficients in a multiple
linear regression model. It is, say, supposed that n > k observations on the response variable are
available: y1, y2,…., yn. Along with each observed response yi, each regressor variable has to be
observed, xij denotes the i-th observation or level of variable xj. The model in terms of the
observations may be written in matrix notation as:
y = Xβ + ε
……………………………………(4.14)
Where,
y is an n x 1 vector of the observations,
X is an n x p matrix of the levels of the independent variables,
β is a p x 1 vector of the regression coefficients, and
ε is an n x 1 vector of random errors.
It is required to find the vector of least squares estimators, b, that minimizes:
L=
…………………….………....(4.15)
ε12 = ε’ε = (y-Xβ)’(y-Xβ)
After some simplifications, the least squares estimator of β is:
b= (x’x) -1 x’y
…………………………………(4.16)
It is easy to see that X’X is a p x p symmetric matrix and X’y is a p x 1 column vector. The
matrix X’X has the special structure. The diagonal elements of X’X are the sums of squares of
33
the elements in the columns of X, and the off-diagonal elements are the sums of cross-products
of the elements in the columns of X. Furthermore, the elements of X’y are the sums of crossproducts of the columns of X and the observations {yi}.
The fitted regression model is:
Ŷ=Xb
…………………………..(4.17)
The difference between the observation yi and the fitted value
ei = yi – Ŷ
is a residual,
…………………………..(4.18)
The n x 1 vector of residuals is denoted by:
e=y–Ŷ
…………………………(4.19)
4.3.3 Model adequacy checking:
It is always necessary to
Examine the fitted model to ensure that it provides an adequate approximation to the
true system.
Verify that none of the least squares regression assumptions are violated.
4.3.4 Properties of the least square estimators:
The method of least squares produces an unbiased estimator of the parameter β in the
multiple linear regression models. The important parameter is the sum of squares of the
residuals
SSE =
(yi-Ŷi )2 =
ei2 = e’e
………………………….……(4.20)
Because X'Xb = X’y, a computational formula for SSE can be derived as:
SSE = y’y – b’X’y
……………………………(4.21)
Equation (4.21) is called the error or residual sum of squares.
34
It can be shown that an unbiased estimator of σ2 is:
…………...……...……..(4.22)
σ2=
Where,
n is a number of observations and
p is a number of regression coefficients.
The total sum of squares is:
SST = y’y –
–
=
....……………….…….(4.23)
Then the coefficient of multiple determination R2 is defined as:
………………….(4.24)
R 2 =1-
R2 is a measure of the amount of reduction in the variability of y obtained by using the regressor
variables x1,x2,...,xk in the model. From inspection of the analysis of variance identity equation
However, a large value of R2 does not necessarily
(Equation (4.24)) can see that
imply that the regression model is good one. Adding a variable to the model will always increase
R2, regardless of whether the additional variable is statistically significant or not. Thus it is
possible for models that have large values of R2 to yield poor predictions of new observations or
estimates of the mean response.
Because R2 always increases as terms are added to the model, it is preferable to use an adjusted
R2 statistic defined as:
R adj2 =
= 1-
(1-R 2)
…………………..……(4.25)
In general, the adjusted R2 statistic will not always increase as variables are added to the
model. In fact, if unnecessary terms are added, the value of R2adj will often decrease. When R2
35
and R2adj differ dramatically, there is a good chance that non significant terms have been
included in the model.
However, testing hypotheses on the individual regression coefficients are very much important.
Such tests would be useful in determining the value of each of the regressor variables in the
regression model. For example, the model might be more effective with the inclusion of
additional variables, or perhaps with the deletion of one or more of the variables already in the
model.
Adding a variable to the regression model always causes the sum of squares for regression to
increase and the error sum of squares to decrease. It must be decided whether the increase in the
regression sum of squares is sufficient to warrant using the additional variable in the model.
Furthermore, adding an unimportant variable to the model can actually increase the mean square
error, thereby decreasing the usefulness of the model [17].
4.3.5 Residual analysis:
The residuals from the least squares fit, defined by
play an important role
in judging model adequacy. It is preferable to work with scaled residuals, in contrast to the
ordinary least squares residuals. These scaled residuals often convey more information than do
the ordinary residuals. The standardizing process scales the residuals by dividing them by their
average standard deviation. In some data sets, residuals may have standard deviations that differ
greatly. There is some other way of scaling that takes this into account. The vector of fitted
values yˆI corresponding to the observed values yi is
Ŷ = Xb = X(X’X) -1 X’y =Hy
36
…………………………….(4.26)
The n x n matrix H = X(X’X)-1 X’ is usually called the hat matrix because it maps the vector of
observed values into a vector of fitted values. The hat matrix and its properties play a central role
in regression analysis.
e = y – Xb =y - Hy =(1-H)y
……………………………… (4.27)
The prediction error sum of squares provides a useful residual scaling:
………………… (4.28)
PRESS =
From Equation (5.28) it is easy to see that the PRESS residual is just the ordinary residual
weighted according to the diagonal elements of the hat matrix hii. Generally, a large difference
between the ordinary residual and the PRESS residual will indicate a point where the model fits
the data well, but a model built without that point predicts poorly.
4.4 Variable Selection and Model Building in Regression:
In response surface analysis, it is customary to fit the full model corresponding to the situation at
hand. It means that in steepest ascent, the full first-order model is usually fitted, and in the
analysis of the second-order model, the full quadratic is usually fitted. Nevertheless, in some
cases, where the full model may not be appropriate; that is, a model based on a subset of the
regressors in the full model may be superior. Variable selection or model-building techniques
usually is used to identify the best subset of regressors to include in a regression model Now, it is
assumed that there are K candidate regressors denoted x1,x2,...,xk and a single response variable
y. All models will have an intercept term β0, so that the full model has (K + 1) parameters. It is
shown that there is a strong motivation for correctly specifying the regression model: Leaving
out important regressors introduces bias into the parameter estimates, while including
unimportant variables weakens the prediction or estimation capability of the model.
37
4.4.1 Procedures for variable selection:
Now, it is required to find several of the more widely used methods for selecting the appropriate
subset of variables for a regression model. The approach is also made on the optimization
procedure used for selecting the best model from the whole set of models and finally it is
required to discuss and illustrate several of the criteria that are typically used to decide which
subset of the candidate regressors leads to the best model.
4.4.2 All possible regression:
This procedure requires that all the regression equations are fitted involving one-candidate
regressors, two-candidate regressors, and so on. These equations are evaluated according to some
suitable criterion, and the best regression model selected. If it is assumed that the intercept term
β0 is included in all equations, then there are K candidate regressors and there are 2 K total
equations to be estimated and examined. For example, if K = 4, then there are 24= 16 possible
equations, whereas if K = 10, then there are210 = 1024. Clearly the number of equations to be
examined increases rapidly as the number of candidate regressors increases. Usually, the
candidate variables are restricted for the model to those in the full quadratic polynomial and it is
required that all models obey the principal of hierarchy. A model is said to be hierarchical if the
presence of higher-order terms (such as interaction and second-order terms) requires the
inclusion of all lower-order terms contained within those of higher order. For example, this
would require the inclusion of both main effects, if a two-factor interaction term was in the
model. Many regression model builders believe that hierarchy is a reasonable model-building
practice while fitting polynomials.
38
4.4.3 Stepwise regression methods:
As the evaluation of all possible regressions can be burdensome, various methods have been
developed for evaluating only a small number of subset regression models by either adding or
deleting regressors one at a time. These methods are generally referred to as stepwise-type
procedures. They can be classified into three broad categories:
a) Forward selection,
b) Backward elimination, and
c) Stepwise regression, which is a popular combination of procedures (a) and (b).
a) Forward selection:
This procedure begins with the assumption that there are no regressors in the model other than
the intercept. An effort is made to find an optimal subset by inserting regressors into the model
one at a time. The first regressor selected for entry into the equation is the one that has the largest
simple correlation with the response variable y. If it is supposed that the first regressor is x1, then
the regressor will produce the largest value of the F-statistic for testing significance of
regression. This regressor is entered, if the F-statistic exceeds a pre-selected F-value, say, Fin (or
F-to-enter). The second regressor chosen for entry is the one that now has the largest correlation
with y after adjusting for the effect of the first regressor entered (x 1) on y. It is referred as partial
correlations. In general, at each step the regressor having the highest partial correlation with y (or
equivalently the largest partial F-statistic given the other regressors already in the model) is
added to the model, if its partial F-statistic exceeds the pre-selected entry level Fin.
The
procedure terminates either when the partial F-statistic at a particular step does not exceed Fin or
when the last candidate regressor is added to the model.
39
b) Backward elimination:
Forward selection begins with no regressors in the model and attempts to insert variables until a
suitable model is obtained. Backward elimination attempts to find a good model by working in
the opposite direction. That is, it is required to start a model that includes all K Candidate
regressors. Then the partial F-statistic (or a t-statistic, which is equivalent) is computed for each
regressor, as if, it were the last variable to enter the model. The smallest of these partial Fstatistics is compared with a pre-selected value, Fout (or F-to-remove); and if the smallest partial
F-value is less than Fout, that regressor is removed from the model. Now, a regression model with
(K – 1) regressors is fitted, the partial F-statistic for this new model calculated, and the procedure
repeated. The backward elimination algorithm terminates, when the smallest partial F-value is
not less than the pre-selected cut-off value Fout.
c) Stepwise regression:
The two procedures described above suggest a number of possible combinations. One of the
most popular is the stepwise regression algorithm. This is a modification of forward selection in
which at each step all regressors, entered into the model previously, are reassessed via their
partial F-or t-statistics. A regressor added at an earlier step may now be redundant because of the
relationship between it and regressors now in the equation. If the partial F-statistic for a variable
is less than Fout, that variable is dropped from the model. Stepwise regression requires two cutoff values, Fin and Fout. Sometimes, it is preferred to choose Fin = Fout, although this is not
necessary. Sometimes, it is also chosen that Fin > Fout, making it more difficult to add a regressor
than to delete one. In the present study, some of the concepts of RSM have been used for
predicting the FDM response viz, Tensile Strength, Flexural strength, and Impact strength. Also,
the optimization of process parameters has been done by RSM.
40
CHAPTER 5
______________________________________________________________________________
SPECIMEN PREPARATION,
EXPERIMENT AND ANALYS
5.1. Specimen preparation
Tensile test specimens having dimensions 60 mm x 20 mm x 5 mm. Flexural test specimens
were 80 mm x 10 mm x 4 mm, and impact test specimens 80 mm x 10 mm x 4 mm, and V notch
of radius 0.25 mm. Since orientation is an important parameter for part strength, tests have been
conducted by changing the orientation for measuring Tensile strength (ASTMD 638), Flexural
strength (ASTMD 790) and Impact strength (ASTM D256). The part are modeled in CATIAV5
software and exported as STL file. STL file is imported to FDM software (Insight). Here, factors
are set as per experiment plan. Three parts per experiment are fabricated using FDM Vantage SE
machine[18]. The material use for part fabrication is ABS P400. Parts are modeled and
experiment is conducted as per ISO R527:1966, ISO 178:1975 and ISO 180:1982 for tensile,
flexural and impact tests respectively. And part made according to response surface design 32
runs.
Table 5.1 Domain of experiments
Level
Factor
Symbol
Unit
Lowest(-2)
Low(-1)
Middle(0) High(1)
Highest(2)
Layer
A
Mm
0.127
0.158
0.190
0.222
0.254
B
degree
0
15
30
45
60
Raster angle
C
degree
0
15
30
45
60
Raster width
D
Mm
0.4064
0.4264
0.4464
0.4664
0.4864
Air gap
E
Mm
0
0.002
0.004
0.006
0.008
thickness
Sample
orientation
41
Figure 5.1 Line diagram of specimen for tensile test
Figure 5.2 Line diagram of specimen for flexural test
Figure 5.3 Line diagram of specimen for izod test
After making the test specimen According to response surface design 32 runs on the FDM
Vantage SE (RP) machine, these specimens were tested. Tensile test and 3-point bending test
42
(flexural test) were conducted on Instron 1195 machine and Impact test was on Instron - Wolpert
Pendulum Impact Testing Machine.
5.2. Testing of specimens:
Tensile strength at break is determined according to ISO R527:1966. Shows the shape of the test
specimens. Flexural strength at yield of test specimen is determined as per ISO R178:1975
standard. Three point bending test is used for flexural strength determination. For this specimen
is supported by two supports and loaded in the middle by force, until the test specimen fractures
[19]. The tensile testing and three-point bending tests were performed using an Instron 1195
series IX automated material testing system with crosshead speeds of 1mm/s and 2mm/s
respectively. Charpy impact test performed in Instron Wolpert pendulum impact test machine is
used to determine the impact strength of specimen as per ISO/179:1982. During impact testing
specimen is subjected to quick and intense blow by hammer pendulum striking the specimen
with a speed of 3.8 m/s. The impact energy absorbed is measure of the toughness of material and
it is calculated by taking the difference in potential energies of initial and final position of
hammer.
following condition were taken during the different test:
Machine parameter of Instron 1195 tensile test:
Sample type
:
ASTM
Sample rate (pts/sec)
:
9.103
Cross head speed (mm/min)
:
1.000
Full scale loading range(KN)
:
20.00
Humidity (%)
:
50%
Temperature (deg.F)
:
73
43
(a) Figure of tensile test
(b) specimen after fracture
Figure 5.4 Instron test of tensile specimen
Machine parameter of Instron 1195 flexural test :
Sample type
:
ASTM
Sample rate (pts/sec)
:
9.103
Cross head speed(mm/min)
:
2.00
Full scale load range (KN)
:
5.00
Humidity (%)
:
50%
Temperature ( deg. F)
:
73
(c) Figure of flexural test
(d) Specimen after bending
Figure 5.5 3-point bending test of specimen
44
Machine parameter Instron - Wolpert Pendulum Impact Testing Machine:
Sample type
:
ASTM
Impact test striking velocity
:
3.8 m/sec
Pendulum fall angle
:
160o
pendulum fall height
:
0.757 m
length of pendulum
:
390 mm
(e) Figure of impact test
(f) specimen after break
Figure 5.6 Impact test of specimen
In order to build empirical model for Tensile strength, flexural strength and impact strength,
experiments were conducted based on central composite design (CCD). The CCD is capable of
fitting second order polynomial and is preferable if curvature is assumed to be present in the
system. To reduce the experiment run, half factorial (two levels) is considered. Maximum and
minimum value of each factor is coded into +2 and -2 respectively using, so that all input factors
are represented in same range [17].
45
xij - (xi,max+xi,min)/2
= 2 x ______________
(xi,max-xi,min)/2
ξij
where
ij
and xij are coded and actual value of j th level of i th factor respectively.
Table 5.2 Experimental data obtained from the CCD runs
Run
Order
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
Factor(coded units)
A
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-2
2
0
0
0
0
0
0
0
0
0
0
0
B
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
0
0
-2
2
0
0
0
0
0
0
0
0
0
C
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
1
1
0
0
0
0
-2
2
0
0
0
0
0
0
0
D
-1
-1
-1
-1
-1
-1
-1
-1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
-2
2
0
0
0
0
0
E
1
-1
-1
1
-1
1
1
-1
-1
1
1
-1
1
-1
-1
1
0
0
0
0
0
0
0
0
-2
2
0
0
0
46
Tensile Flexural Impact
strength strength strength
(MPa)
(MPa) (Joules)
11.54
31.70
11.80
17.76
31.40
11.30
11.04
25.00
10.50
13.69
26.50
14.00
12.29
31.90
10.85
12.35
42.00
11.35
11.15
32.10
12.85
12.29
38.80
11.60
16.73
30.10
10.79
16.17
31.60
13.80
11.04
29.70
10.90
11.86
19.20
11.90
12.94
31.80
12.80
15.60
31.90
10.40
11.05
35.70
11.70
16.31
34.00
11.37
11.14
39.70
11.50
16.10
39.10
12.14
16.55
32.00
10.80
11.04
24.20
12.30
15.60
21.30
13.40
12.30
34.20
12.30
12.26
37.50
11.40
16.40
36.00
12.40
13.98
39.70
11.50
12.40
39.80
14.30
15.20
42.20
14.90
16.30
40.10
15.50
15.90
41.60
14.70
30
31
32
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
16.40
15.50
15.90
41.10
42.30
41.80
15.00
15.10
14.90
5.3 Analysis of experiments:
Analysis of the experimental data obtained from CCD design runs is done on MINITAB R14
software using full quadratic response surface model as given by .
y
k
0
i 1
i xi
k
xx
i 1
ii i i
i j
x xj
ij i
Where y is the response, xi is i th factor
For significance check F value given in ANOVA table is used. Probability of F value greater
than calculated F value due to noise is indicated by p value. If p value is less than 0.05,
significance of corresponding term is established. For lack of fit p value must be greater the 0.05.
An insignificant lack of fit is desirable as it indicates anything left out of model is not significant
and develop model fits.
Based on analysis of variance (ANOVA) test full quadratic model was found to be suitable for
tensile strength, flexural strength and impact strength with regression p-value less than 0.05 and
lack of fit more then 0.05.
5.3.1 Analysis of experiment for tensile test:
Table 5.3 Estimated regression coefficients for tensile test
Term
Constant
A
B
C
D
E
A*A
B*B
C*C
Coef.
15.8469
1.1737
-1.1654
-0.5188
-0.7446
-0.2746
-0.5419
-0.4982
-0.4594
SE Coef.
0.2408
0.1232
0.1232
0.1232
0.1232
0.1232
0.1115
0.1115
0.1115
47
T
65.814
9.525
-9.458
-4.210
6.042
-2.228
-4.862
-4.470
-4.122
P
0.000
0.000
0.000
0.001
0.000
0.048
0.001
0.001
0.002
D*D
E*E
A*B
A*C
A*D
A*E
B*C
B*D
B*E
C*D
C*E
D*E
-0.3644
-0.6494
0.0931
-0.0006
-0.1181
0.3406
0.7619
-0.3381
0.9581
0.3781
0.4044
0.3669
S =0.6037
0.1115
0.1115
0.1509
0.1509
0.1509
0.1509
0.1509
0.1509
0.1509
0.1509
0.1509
0.1509
-3.270
-5.826
0.617
-0.004
-0.783
2.257
5.048
-2.240
6.349
2.505
2.679
2.431
R2 = 97.4%
0.007
0.000
0.550
0.997
0.450
0.045
0.000
0.047
0.000
0.029
0.021
0.033
R2 (adj) = 92.7%
In tensile test analysis all the factors and interaction A*A, B*B, C*C, D*D, E*E, A*E, B*C,
B*D, B*E, C*D, C*E, and D*E interactions are important because their P value is less than 0.05.
The coefficient of determination (R2) which indicates the goodness of fit for the model so the
value of R2 =97.4% which indicate the high significance of the model.
With the above analysis we found the following regression equation:Ts = 15.8469+1.1737A -1.1654B -0.5188C +0.7446D -0.2746E -0.5419(A*A) - 0.4982(B*B) 0.4594(C*C) -0.3644(D*D) - 0.6494(E*E) + 0.3406(A*E) +0.7619(B*C) – 0.3381(B*D) +
0.9581(B*E) + 0.3781(C*D) + 0.404(C*E) + 0.3669(D*E).
Table 5.4 Analysis of variance for tensile test
Source
DF
Regression
20
Linear
5
Square
5
Interaction
10
Residual
11
Lack of fit
6
Pure error
5
Total
31
DF=degree of freedom
Tensile strength
MS
7.5761
17.4470
5.8413
3.5081
0.3644
0.4926
0.2107
SS
151.522
87.235
29.206
35.081
4.009
2.955
1.053
155.530
SS= sum of square
48
F
20.79
47.87
16.03
9.63
p
0.000
0.000
0.000
0.000
2.34
0.185
MS=mean sum of square
In the above table we can see P value of all the term is less than 0.05, so these all term are
significant, and non significance lack of fit is desired in this case value of LOF is 0.185 which is
more than 0.05 and non significant.
5.3.2 Response surface analysis for tensile test:
Surface Plot of tensile vs B, A
Surface Plot of tensile vs C, A
Hold Values
C 0
D 0
E 0
Hold Values
B 0
D 0
E 0
17.5
15
tensile
15.0
tensile
12
9
10.0
2
6
0
-2
A
0
2
12.5
B
2
0
-2
-2
A
Surface Plot of tensile vs D, A
0
2
C
-2
Surface Plot of tensile vs E, A
Hold Values
B 0
C 0
E 0
Hold Values
B 0
C 0
D 0
18
15
15
tensile
tensile
12
12
9
9
2
0
-2
A
0
2
2
6
D
0
-2
-2
A
49
0
2
-2
E
Surface Plot of tensile vs C, B
Surface Plot of tensile vs D, B
Hold Values
A 0
D 0
E 0
Hold Values
A 0
C 0
E 0
20
tensile
17.5
15.0
15
tensile
12.5
10
2
0
-2
B
0
0
-2
-2
2
2
10.0
C
B
Surface Plot of tensile vs E, B
0
2
D
-2
Surface Plot of tensile vs D, C
Hold Values
A 0
C 0
D 0
Hold Values
A 0
B 0
E 0
16
15
14
tensile
tensile
10
10
2
5
0
-2
0
B
12
E
-2
2
2
0
-2
C
Surface Plot of tensile vs E, C
0
D
-2
2
Surface Plot of tensile vs E, D
Hold Values
A 0
B 0
D 0
Hold Values
A 0
B 0
C 0
16
15.0
14
tensile
tensile
12.5
12
10.0
10
2
0
-2
C
0
2
2
E
0
-2
-2
D
0
2
Figures 5.7 Surface plots for tensile strength
50
-2
E
From response surface it is observed that strength first decrease and then increase with layer
thickness (A) increase. The reason may be that at smaller layer thickness numbers of layers are
more resulting in increase in heat conduction towards the bottom layers therefore strong bonding
between adjacent rasters is expected. But this also increases the distortion in bottom layers which
is responsible for weak bond strength, so with increase in layer thickness distortion in layer
thickness decreased and tensile strength increased.
Response surface shows the decreasing trend of strength with respect to increase in orientation
(B).this may be due to the stepped effect in which one layer does not coincide with the next layer
exactly this ultimately reduce the strength of the part. Number of layers also increases with
increase in orientation for same layer thickness as a result distortion in part will increase
resulting in less bond strength.
Tensile strength is decreasing with respect to raster angle ,the reason may be that at small angle
raster deposited have long length due to this inter bonding of the part is good and also number of
layer is less. So, with increase in raster angle this inters bonding gets weak and decreases the
strength.
Tensile strength is increase with increasing the raster width (D) , the reason may be that at small
raster width number of layer are more so the distortion chances are higher, so with increase in
raster width strength increase.
Tensile strength is increased up to certain level and after that level it decrease, this is due to that
If raster deposition is much closed the melt material can overlap to next raster and it can lose
Its strength. But this air gap should be maintain, if air gap is large there will be a gap between
two raster and it can also reduce the strength.
51
(a) Horizontal position of specimen
(b) vertical position of specimen
Figure 5.8 SEM image of tensile failure of specimen
The above SEM images are shown for two position of tensile fracture (a) is a horizontal position of
fractured specimen and (b) is vertical position of fractured specimen. We have 32 specimens for each
test, so randomly we have chosen one specimen of parametric combination, and choose the specimen
of parametric combination of 19 for the SEM image which will give the behavior of specimen after
different test.
In Image (a) the circle portion shows the fracture on the specimen, This is the ABS P400 (acrylonitrilebutadine-styrene) material, and we can see there is no or negligible elongation in the specimen while we
do the tensile test and material behavior like a brittle material and image (b) shows the different points
from where fracture occurred.
52
5.3.3 Analysis of experiment for flexural test:
Table 5.5 Estimated regression coefficients for flexural test
Term
Constant
A
B
C
D
E
A*A
B*B
C*C
D*D
E*E
A*B
A*C
A*D
A*E
B*C
B*D
B*E
C*D
C*E
D*E
S = 1.320
Coef.
41.7159
0.2583
-1.5417
3.2833
-0.7667
0.6500
-0.7284
-3.5534
-3.6409
-1.3909
-0.6409
-0.9625
1.4375
-1.7875
0.6375
1.7125
0.4875
-0.5125
-0.4625
-0.7625
0.3125
SE Coef.
0.5264
0.2694
0.2694
0.2694
0.2694
0.2694
0.2437
0.2437
0.2437
0.2437
0.2437
0.3299
0.3299
0.3299
0.3299
0.3299
0.3299
0.3299
0.3299
0.3299
0.3299
T
79.248
0.959
-5.723
12.188
-2.846
2.413
-2.989
-14.583
-14.942
-5.708
-2.630
-2.917
4.357
-5.418
1.932
5.190
1.478
-1.553
-1.402
-2.311
0.947
R2 = 98.5%
P
0.000
0.358
0.000
0.000
0.016
0.034
0.012
0.000
0.000
0.000
0.023
0.014
0.001
0.000
0.079
0.000
0.168
0.149
0.189
0.041
0.364
R2(adj) = 95.7%
In flexural test we can see by the above table factors B, C, D, E and interaction A*A, B*B, C*C,
D*D, E*E, A*B, A*C, A*D, B*C, and C*E are most important because they are having P value
less than 0.05.
The coefficient of determination (R2) which indicates the goodness of fit for the model so the
value of R2 =98.5% which indicate the high significance of the model.
With the above analysis we found the following regression equation:-
53
Fs = 41.7159– 1.5417B + 3.2833C – 0.7667D + 0.6500E – 0.7284(A*A) – 3.5534 (B*B) –
3.6409 (C*C) – 1.3909 (D*D) – 0.6409 (E*E) – 0.9625 (A*B) + 1.4375(A*C) -1.7875 (A*D) +
1.7125(B*C) – 0.7625(C*E).
Table 5.6 ANOVA analysis for flexural test
Source
Regression
Linear
Square
Interaction
Residual
Lack of fit
Pure error
Total
DF
20
5
5
10
11
6
5
31
Flexural strength
MS
F
61.922
35.55
68.323
39.23
144.420
82.92
17.473
10.03
1.742
2.635
3.94
0.670
SS
1238.44
341.62
722.10
174.73
19.16
15.81
3.35
1257.60
DF =degree of freedom
SS = sum of square
p
0.000
0.000
0.000
0.000
0.077
MS = mean sum of square
In the above table we can see P value of all the term is less than 0.05, so these all term are
significant, and non significance lack of fit is desired in this case value of LOF is 0.077 which is
more than 0.05 and non significant.
5.3.4 Response surface analysis for flexural test:
Surface Plot of flexural vs B, A
Surface Plot of flexural vs C, A
Hold Values
C 0
D 0
E 0
Hold Values
B 0
D 0
E 0
40
flexural
40
flexural
30
30
20
2
20
0
-2
A
0
2
2
10
B
0
-2
-2
A
54
0
2
-2
C
Surface Plot of flexural vs D, A
Surface Plot of flexural vs E, A
Hold Values
B 0
C 0
E 0
Hold Values
B 0
C 0
D 0
42
40
39
35
flexural
flexural
36
30
2
25
0
-2
0
A
0
-2
-2
2
2
33
D
0
A
2
E
-2
Surface Plot of flexural vs D, B
Surface Plot of flexural vs C, B
Hold Values
A 0
C 0
E 0
Hold Values
A 0
D 0
E 0
45
40
30
flexur al
flexur al
15
2
0
0
-2
0
B
2
30
2
20
C
0
-2
-2
B
Surface Plot of flexural vs E, B
0
D
-2
2
Surface Plot of flexural vs D, C
Hold Values
A 0
C 0
D 0
Hold Values
A 0
B 0
E 0
40
flexural
40
flexural
30
20
2
20
0
-2
B
0
2
30
E
2
0
-2
-2
C
55
0
2
-2
D
Surface Plot of flexural vs E, D
Surface Plot of flexural vs E, C
Hold Values
A 0
B 0
C 0
Hold Values
A 0
B 0
D 0
40
flexural
40
30
flexural
35
20
2
10
0
-2
C
0
2
2
30
E
0
-2
-2
D
0
2
E
-2
Figure 5.9 Surface plots of flexural test
Flexural strength response surface shows that flexural strength monotonously increases with
increase in layer thickness (A) .The reason may be that for flexural strength measurement load is
applied perpendicular to length of specimen. Therefore thicker part will show more strength as
compared to thinner part. Hence, if part is made with thicker layer then each individual layer will
show more resistance against the failure as compared to part of same thickness made with
thinner layer.
Flexural strength of the material is first increasing with respect to sample orientation (B) and
after certain level it gets decrease, this is may be due to the stepped effect, in this layer deposited
over another layer so due the sample orientation there may be some portion is vacant this lead to
the weak strength of material.
At lower raster angle (C) not only the length of individual raster is more but their inclination
along the length of specimen is also more as result strength will be more. Combine effect of long
raster and thick layer is responsible for monotonous increase in strength at small raster angle
value with increase in layer thickness. But long rasters have more distortion as compare to short
rasters as a result strength will increase on increasing the raster angle.
56
Higher width (D) reducing the number of rasters due to which the rasters are not able to give
much resistance for the applied load. Higher raster not only reduces the number of rasters but
also inputting more heat into the system as a result strength will decrease.
Flexural strength is increased with increasing the air gap (E), this is due to that, at min. air gap
two raster will be very close and this decreases the heat dissipation and increase the chance of
residual stress accumulation. This lead to reduce the strength at low level of air gap.
(c) Crack formation in flexural specimen
(d) specimen after cracking
Figure 5.10 SEM image of crack surface of flexural specimen
SEM image of 3-point bending test specimen is shown in Image (c) and (d) respectively. Image
(c) shows the crack occurred after bending test, in fig we can see that the cracking length is
max, where max. Load is applied .and in Image (d) the horizontal position of the cracked
specimen is shown, although ABS P400 (acrylonitrile-butadine-styrene) is a brittle material so
we can see in the Image (d) no or negligible elongation took place.
57
5.3.5 Analysis of experiment for impact test:
Table 5.7 Estimated regression coefficients for impact test
Term
Constant
A
B
C
D
E
A*A
B*B
C*C
D*D
E*E
A*B
A*C
A*D
A*E
B*C
B*D
B*E
C*D
C*E
D*E
S = 0.3835
Coef.
14.9724
0.2004
0.1971
-0.1779
0.0588
0.6429
-0.7549
-0.8224
-0.4974
-0.7349
-0.4849
0.1444
-0.6556
-0.0606
0.0506
0.1569
-0.3481
-0.1869
-0.0106
-0.1369
-0.1044
SE Coef.
0.15296
0.07828
0.07828
0.07828
0.07828
0.07828
0.07081
0.07081
0.07081
0.07081
0.07081
0.09587
0.09587
0.09587
0.09587
0.09587
0.09587
0.09587
0.09587
0.09587
0.09587
T
P
97.884
2.560
2.518
-2.273
0.751
8.213
-10.661
-11.614
-7.025
-10.379
-6.848
1.506
-6.838
-0.632
0.528
1.636
-3.631
-1.949
-0.111
-1.428
-1.089
R2 = 97.9%
0.000
0.027
0.029
0.044
0.469
0.000
0.000
0.000
0.000
0.000
0.000
0.160
0.000
0.540
0.608
0.130
0.004
0.077
0.914
0.181
0.300
R2 (adj) =94.0%
With the above analysis we found factors A, B, C, E and interaction A*A, B*B, C*C, D*D, E*E,
A*C, B*D.
The coefficient of determination (R2) which indicates the goodness of fit for the model so the
value of R2 =97.9% which indicate the high significance of the model.
With the above analysis we found the following regression equation:Is = 14.9724 + 0.2004A + 0.1971B – 0.1779C + 0.6429E – 0.7549(A*A) – 0.8224(B*B) 0.4974 (C*C) – 0.7349 (D*D) - 0.4849 (E*E) – 0.6556 (A*C) – 0.3481(B*D) .
58
Table 5.8 Analysis of variance for impact test
Source
DF
Regression
20
Linear
5
Square
5
Interaction
10
Residual
11
Lack of fit
6
Pure error
5
Total
31
DF =degree of freedom
SS
74.2303
12.6590
50.8931
10.6783
1.6177
1.2494
0.3683
75.8480
SS = sum of square
Tensile strength
MS
3.7115
2.5318
10.1786
1.0678
0.1471
0.2082
0.0737
F
25.24
17.22
69.21
7.26
p
0.000
0.000
0.000
0.001
2.83
0.137
MS = mean sum of square
In the above table we can see P value of all the term is less than 0.05, so these all term are
significant, and non significance lack of fit is desired in this case value of LOF is 0.077 which is
more than 0.05 and non significant.
5.3.6 Response analysis for impact test:
Surface Plot of impact vs B, A
Surface Plot of impact vs C, A
Hold Values
C 0
D 0
E 0
Hold Values
B 0
D 0
E 0
14
impact
14
12
impact
12
10
10
2
8
0
-2
A
0
2
2
8
B
0
-2
-2
A
59
0
2
-2
C
Surface Plot of impact vs D, A
Surface Plot of impact vs E, A
Hold Values
B 0
C 0
E 0
Hold Values
B 0
C 0
D 0
16
14
impact
14
12
impact
10
10
2
8
0
-2
A
0
12
D
-2
2
2
0
-2
0
A
Surface Plot of impact vs C, B
E
-2
2
Surface Plot of impact vs D, B
Hold Values
A 0
D 0
E 0
Hold Values
A 0
C 0
E 0
14
14
impact
12
impact
12
10
10
8
2
8
0
-2
0
B
C
-2
2
2
0
-2
B
Surface Plot of impact vs E, B
0
D
-2
2
Surface Plot of impact vs D, C
Hold Values
A 0
C 0
D 0
Hold Values
A 0
B 0
E 0
14
impact
14
12
impact
10
2
8
0
-2
B
0
2
12
2
10
E
0
-2
-2
C
60
0
2
-2
D
Surface Plot of impact vs E, C
Surface Plot of impact vs E, D
Hold Values
A 0
B 0
D 0
Hold Values
A 0
B 0
C 0
16
14
14
impact
impact
12
12
10
2
10
0
-2
C
0
2
2
8
E
0
-2
-2
D
0
2
E
-2
Figure 5.11 Surface plots of impact test
With the above graphs we can see that behavior of all the factors are similar and up to level 0 of
the factors strength is increasing and after that level its decreasing, this is may be due to the
reason which we have discussed in tensile strength analysis and flexural strength analysis.
(e) Horizontal position of broke specimen
(f) vertical position of broke specimen
Figure 5.12 SEM image of broke impact test specimen
61
In above SEM image two image shown (e) and (f) respectively for horizontal and vertical
position of the impact specimen. Image (e) shows the behavior of material after breaking. In this
test material directly break from the notch portion with the help of pendulum. And it is the ABS
P400 (acrylonitrile-butadine-styrene) material which behaves as a brittle material, this directly
break and no elongation takes place, this we can see in image (e) and (d) that the material broke
at different place due to sudden impact load, but no change or elongation in any dimension of the
shape took place.
With the above three analysis we see that tensile strength is higher in experiment no. 2, flexural
strength is higher in experiment no. 31, and impact strength is higher in experiment no. 28. And
factors and interactions which are affecting the strength are different in three cases. So to
optimize all the response simultaneously by converting multiple response into single response
Gery-taguchi method is used.
5.4 Optimization of process parameters
Above discussion shows that FDM process involves large number of conflicting factors and complex
part building phenomena making it difficult to predict the output characteristics based on simple analysis
of factor variation. Hence, to determine the optimal setting of process parameters that will
maximize the tensile strength, flexural strength and impact strength respectively, desirability
function (DF) given by Equation (5.1) is used
n
1/
n
DF
d
i 1
wi
i
wi
i 1
………………..……………………(5.1)
where di is the desirability defined for the ith targeted output. For a goal to find a maximum, di is
calculated as shown in Equation (5.2).
62
di
0
Yi
if
Low i
Yi low i
High i low i
di
di
if
1
if
Yi
low i
Yi
High i
……………....(5.2)
High i
where Yi is the found value of the ith output during optimization process, and the lowi, Highi are
the minimum and maximum values respectively of the experimental data for the ith output. Since
all the strengths are equally important therefore value of weight wi is taken as 1.Optimum factor
levels that will maximize the desirability function are calculated and are given in Table (5.10) for
respective strength together with its predicted value.
Table 5.9 Optimum factor and predicted response for individuals strength
Response
Goal
Tensile strength
Low
High
wi
Maximum
11.04 17.76
1
Flexural strength
Maximum
19.2
42.3
1
Impact strength
Maximum
10.4
15.5
1
63
Factor level
(Coded units)
A=2; B=-2; C=2; D= -2; E=-2
A=0; B=0; C=0;
D=0; E=0.4968
A=0; B=0; C=0;
D=0; E=0.6835
Predicted
response
19.43
DF
1.0000
41.88
0.98185
15.1853
0.93829
CHAPTER 6
______________________________________________________________________________
GREY – BASED TAGUCHI METHOD
6. Grey-based taguchi method
6 .1 Introduction :
Taguchi’s philosophy, developed by Dr. Genichi Taguchi, is an efficient tool for the design of
high quality manufacturing system [30]. It is a method based on Orthogonal Array experiments
which provides much-reduced variance for the experiment resulting optimal setting of process
control parameters. Orthogonal Array provides a set of well-balanced experiments with less
number of experimental runs. In order to evaluate the optimal parameter setting, Taguchi method
uses a statistical measure of performance called signal-to-noise ratio that takes both the mean and
the variability into account. The S/N ratio is the ratio of the mean (signal) to the standard
deviation (noise). The ratio depends on the quality characteristics of the product/process to be
optimized. The standard S/N ratios generally used are Nominal-is-Best (NB), lower-the-better
(LB) and Higher-the-Better (HB). The optimal setting is the parametric combination, which has
the highest S/N ratio. However, traditional Taguchi method cannot solve multi-objective
optimization problem. This can be achieved by grey based Taguchi method. In grey relational
analysis, experimental data i.e. measured features of quality characteristics of the product are
first normalized ranging from zero to one. This process is known as grey relational generation.
Next, based on normalized experimental data, grey relational coefficient is calculated to
represent the correlation between the desired and actual experimental data. Then, overall grey
relational grade is determined by averaging the grey relational coefficient corresponding to
selected responses. The overall performance characteristic of the multiple response process
depends on the calculated overall grey relational grade. This approach converts a multipleresponse process optimization problem into a single response optimization situation with the
objective function is overall grey relational grade. Using grey-Taguchi method, the optimal
parametric combination is then evaluated by maximizing the S/N ratio of the overall grey
relational grade.
64
6.2 Grey-relational analysis:
Data preprocessing:
Grey data processing must be performed before Grey correlation coefficients can be calculated.
A series of various units must be transformed to be dimensionless. Usually, each series is
normalized by dividing the data in the original series by their average. Let the original reference
sequence and sequence for comparison be represented as xo(k) and xi (k), i =1, 2, ...,m; k =1, 2,
..., n, respectively, where m is the total number of experiment to be considered, and n is the total
number of observation data. Data preprocessing converts the original sequence to a comparable
sequence [30]. Several methodologies of preprocessing data can be used in Grey relation
analysis, depending on the characteristics of the original sequence. If the target value of the
original sequence is “the-larger-the-better”, then the original sequence is normalized as follows:
If the expectancy is smaller- the –better, then the original sequence should be normalized as follow.
However, there is a definite target value to be achieved; the original sequence will be normalized
in the form.
65
or the original sequence can be simply normalized by the most basic methodology, i.e. let the
values of original sequence be divided by the first value of sequence:
where x*i(k) is the value after the grey relational generation (data pre-processing), max x0i (k) is
the largest value of x0i (k), min x0i (k) is the smallest value of x0i (k) and x0 is the desired value.
Grey relational coefficient and grey relational grade:
Following data pre-processing, a grey relational coefficient is calculated to express the
relationship between the ideal and actual normalized experimental results. The grey relational
coefficient can be expressed as follows:
where Δ0i(k)is the deviation sequence of the reference sequence x*0(k) and the comparability sequence
x*i(k), namely
ξ is distinguishing or identification coefficient: ξ ε [0,1]. ξ =0.5 is generally used.
After obtaining the grey relational coefficient, we normally take the average of the grey
relational coefficient as the grey relational grade. The grey relational grade is defined as follows:
66
However, since in real application the effect of each factor on the system is not exactly same. Eq.
(6) can be modified as follow:
Where wk represents the normalized weighting value of factor k. Given the same weights, Eqs.
(6) and (7) are equal. In the grey relational analysis, the grey relational grade is used to show the
relationship among the sequences. If the two sequences are identical, then the value of grey
relational grade is equal to 1. The grey relational grade also indicates the degree of influence that
the comparability sequence could exert over the reference sequence. There- fore, if a particular
comparability sequence is more important than the other comparability sequences to the
reference sequence, then the grey relational grade for that comparability sequence and reference
sequence will be higher than other grey relational grades [31] .
In this analysis we are considering three responses tensile strength, flexural strength, impact
strength, these response should be high, so we normalizing the data according to larger -thebetter(LB).
Table 6.1 . Normalization of the data (larger-the-better) (x*I (k))
Experiment no.
Ideal condition
1
2
3
4
5
6
7
Tensile strength
1.00000
Flexural strength
Impact strength
1.00000
1.00000
0.07440
1.00000
0.00000
0.39435
0.18601
0.19494
0.01637
0.54113
0.52814
0.25108
0.31602
0.54978
0.98701
0.55844
67
0.27451
0.17647
0.01961
0.70588
0.08824
0.18627
0.48039
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
0.18601
0.84673
0.76339
0.00000
0.12202
0.28274
0.67857
0.00149
0.78423
0.01488
0.75298
0.81994
0.00000
0.67857
0.18750
0.18155
0.79762
0.43750
0.20238
0.61905
0.78274
0.72321
0.79762
0.66369
0.72321
0.84848
0.47186
0.53680
0.45455
0.00000
0.54545
0.54978
0.71429
0.64069
0.88745
0.86147
0.55411
0.21645
0.09091
0.64935
0.79221
0.72727
0.88745
0.89177
0.99567
0.90476
0.96970
0.94805
1.00000
0.97835
0.23529
0.07647
0.66667
0.09804
0.29412
0.47059
0.00000
0.25490
0.19020
0.21569
0.34118
0.07843
0.37255
0.58824
0.37255
0.19608
0.39216
0.21569
0.76471
0.88235
1.00000
0.84314
0.90196
0.92157
0.88235
Table 6.2 The deviation sequence (Δo,i(k))
Experiment no.
1
2
3
4
5
6
Tensile strength,
Flexural strength,
Impact strength,
Δo,i(1)
Δo,i(2)
Δo,i(3)
0.92560
0.45887
0.72549
0.00000
0.47186
0.82353
1.00000
0.74892
0.98039
0.60565
0.68398
0.29412
0.81399
0.45022
0.91176
0.80506
0.01299
0.81373
68
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
0.98363
0.81399
0.15327
0.23661
1.00000
0.87798
0.71726
0.32143
0.99851
0.21577
0.98512
0.24702
0.18006
1.00000
0.32143
0.81250
0.81845
0.20238
0.56250
0.79762
0.38095
0.21726
0.27679
0.20238
0.33631
0.27679
0.44156
0.15152
0.52814
0.46320
0.54545
1.00000
0.45455
0.45022
0.28571
0.35931
0.11255
0.13853
0.44589
0.78355
0.90909
0.35065
0.20779
0.27273
0.11255
0.10823
0.00433
0.09524
0.03030
0.05195
0.00000
0.02165
0.51961
0.76471
0.92353
0.33333
0.90196
0.70588
0.52941
1.00000
0.74510
0.80980
0.78431
0.65882
0.92157
0.62745
0.41176
0.62745
0.80392
0.60784
0.78431
0.23529
0.11765
0.00000
0.15686
0.09804
0.07843
0.11765
Table 6.3 Calculation of grey relational coefficients(ξi (k))
Tensile strength
Flexural
Experimental no. (MPa)
strength(MPa)
Impact strength(joules)
1
0.35073
0.52145
0.40800
2
1.00000
0.51448
0.37778
3
0.33333
0.40035
0.33775
4
0.45222
0.42230
0.62963
5
0.38052
0.52619
0.35417
6
0.38312
0.97468
0.38000
7
0.33701
0.53103
0.49038
8
0.38052
0.76744
0.39535
69
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
0.76538
0.67879
0.33333
0.36285
0.41076
0.60869
0.33366
0.69855
0.33667
0.66933
0.73523
0.33333
0.60869
0.38095
0.37923
0.71187
0.47059
0.38532
0.56757
0.69710
0.64367
0.71187
0.59786
0.64367
0.48632
0.51910
0.47826
0.33333
0.52381
0.52619
0.63637
0.58186
0.81626
0.78305
0.52860
0.38954
0.35484
0.58779
0.70642
0.64706
0.81626
0.82206
0.99141
0.84000
0.94286
0.90588
1.00000
0.95850
Table 6.4 Grey- relational grade (γi)
γi
0.42672
0.63075
0.35714
0.50138
0.42029
0.57926
0.45280
0.51443
0.53431
0.59929
0.38941
Ex. No.
1
2
3
4
5
6
7
8
9
10
11
70
0.35124
0.60000
0.35664
0.41463
0.48572
0.33333
0.40157
0.38174
0.38931
0.43147
0.35172
0.44348
0.54839
0.44348
0.38346
0.45133
0.38931
0.68000
0.80952
1.00000
0.76120
0.83606
0.86441
0.80952
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
0.37027
0.47343
0.48940
0.45720
0.55405
0.51408
0.62795
0.53851
0.38878
0.50397
0.47074
0.48970
0.60342
0.55872
0.62912
0.78950
0.84570
0.78257
0.81793
0.82075
0.80389
71
Figure 6.1. Sensitivity analysis for different distinguishing coefficients(ξ)
Figure 6.2 . Grey relational grade variation with number of experiment(ξ=0.5)
72
The purpose of distinguish coefficient is to expand or compress the range of the grey relational
coefficient. The distinguishing coefficient can be selected by decision maker judgement, and
different distinguishing coefficients usually provide different results in GRA. Sensitivity analysis
for different distinguishing coefficients (Figure 19) shows that impact of their variation on grey
relation coefficient is very small. They all led to the same optimum factor levels. In this case
distinguishing coefficient is taken as 0.5.
Table 6.5 Response surface analysis for grey relational grade
S.N.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
A
B
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-2
2
0
0
0
0
0
0
0
0
C
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
0
0
-2
2
0
0
0
0
0
0
D
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
1
1
0
0
0
0
-2
2
0
0
0
0
E
-1
-1
-1
-1
-1
-1
-1
-1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
-2
2
0
0
73
1
-1
-1
1
-1
1
1
-1
-1
1
1
-1
1
-1
-1
1
0
0
0
0
0
0
0
0
-2
2
Grey-relational
grade
0.42672
0.63075
0.35714
0.50138
0.42029
0.57926
0.45280
0.51443
0.53431
0.59929
0.38941
0.37027
0.47343
0.48940
0.45720
0.55405
0.51408
0.62795
0.53851
0.38878
0.50397
0.47074
0.48970
0.60342
0.55872
0.62912
27
28
29
30
31
32
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0.78950
0.84570
0.78257
0.81793
0.82075
0.80389
Table 6.6 Estimated Regression coefficients for grey relational grade
Term
Constant
A
B
C
D
E
A*A
B*B
C*C
D*D
E*E
A*B
A*C
A*D
A*E
B*C
B*D
B*E
C*D
C*E
D*E
S= 0.03359
Coef.
0.806193
0.039803
-0.035677
0.002714
0.008833
0.014307
-0.055893
-0.082735
-0.076808
-0.062007
-0.050166
-0.010024
-0.003793
-0.025638
0.015982
0.036812
-0.005890
0.012165
0.001875
0.009618
0.007965
SE Coef.
0.013399
0.006857
0.006857
0.006857
0.006857
0.006857
0.006206
0.006206
0.006206
0.006206
0.006206
0.008398
0.008398
0.008398
0.008398
0.008398
0.008398
0.008398
0.008398
0.008398
0.008398
R-sq= 98.0%
T
60.170
5.805
-5.203
0.396
1.288
2.087
-9.012
-13.339
-12.384
-9.997
-8.088
-1.194
-0.452
-3.053
1.903
4.383
-0.701
1.449
0.223
1.145
0.948
P
0.000
0.000
0.000
0.700
0.224
0.061
0.000
0.000
0.000
0.000
0.000
0.258
0.660
0.011
0.084
0.001
0.498
0.175
0.827
0.276
0.363
R-sq (adj) = 94.4%
With the above analysis we found while considering all the response simultaneously only factor
(A), and (B), and interaction term A*A, B*B, C*C, D*D, E*E, A*D, B*C.
The coefficient of determination (R2) which indicates the percentage of total variation in the
response explained by the terms in the model is 98%.
74
With the above analysis we found the following regression equation:Os =0.806193+0.039803A – 0.035677B -0.055893(A*A) – 0.082735(B*B) – 0.076808(C*C) 0.062007 (D*D) - 0.050166(E*E) – 0.025638(A*D)+0.036812(B*C) .
Table 6.7 ANOVA analysis for grey relational grade
Source
Regression
Linear
Square
Interaction
Residual
Lack of fit
Pure error
Total
DF
20
5
5
10
11
6
5
31
Tensile strength
MS
0.030792
0.015107
0.099341
0.004360
0.001128
0.001625
0.000533
SS
0.615837
0.075534
0.496706
0.043597
0.012413
0.009750
0.002663
0.628250
F
27.29
13.39
88.03
3.86
p
0.000
0.000
0.000
0.018
3.05
0.121
6.3 Response Optimization of GRG and optimal parameter setting:
With the help of response optimizer we have found the optimal parameter setting for all three
responses:
response
Goal
Maximum
Lower
0.35714
Predicted GRG Responses
desirability
Target
1
Upper
1
Weight
1
= 0.80619
= 0.80619
For the above predicted response the optimal parameter setting is
Parameter
Layer thickness
Sample orientation
Raster angle
Raster width
Air gap
value
0.190
30
30
0.4464
0.004
75
Units
(mm)
(degree)
(degree)
(mm)
(mm)
Importance
1
Table 6.8 Comparison of parameter setting individual and simultaneous optimization
( Uncoded Value)
Parameter
Individual optimization
Tensile
Flexural
Impact
strength
strength
strength
0.254
0.190
0.190
A (mm)
0
30
30
B (degree)
0
30
30
C (degree)
0.4064
0.4464
0.4464
D (mm)
0
0.004993 0.005367
E (mm)
76
Simultaneous optimization
0.190
30
30
0.4464
0.004
CHAPTER 7
______________________________________________________________________________
RESULT, CONCLUSION AND FUTURE SCOPE
7. Results, conclusions and future scope
7.1 Results and discussions:
The tensile strength data of ABS sample with different level of process parameter are shown in
table (2). The ultimate strength were the highest (17.76 MPa) for the layer thickness 0.222 mm,
sample orientation 15o , raster angle 15o , raster width 0.4264 mm, and air gap 0.002 mm
combination set and the lower strength (11.04 MPa) for layer thickness 0.158 mm, sample
orientation 45o, raster angle 15o , raster width 0.4264 mm, and air gap 0.002 mm, the lower
strength in the solid model could have been caused by residual stress from the volumetric
shrinkage , weak interlayer bonding, or interlayer porosity. Examination of the fracture surfaces
revealed fracture paths that were controlled by either weak interlayer bonding or interlayer
porosity. Weak interlayer bonding probably was a result of residual stresses caused by
volumetric shrinkage of the polymer layers during solidification from the melt. Weak interlayer
bonding could also be caused by the low molecular diffusion and low cross-linking between the
polymer layers during deposition from the melt. In addition, the interlayer porosity reduced the
load-bearing area across the layers and hence provided an easy fracture path. The percent
elongation of the tensile specimens was <2%, and the ABS material failed in a semi-brittle
manner. Because the ABS elongation was so low.
The 3-point bending test data of ABS sample shown in table (2).the flexural strength are greater
than the tensile strengths because the modulus of rupture measure the maximum strength at the
outer fiber of the beam. This is expected because during bending, the sample is subjected to both
compressive and tensile load in this test flexural strength is highest (42.3 MPa) for the layer
thickness 0.190 mm, sample orientation 30o , raster angle 30o, raster width 0.4464 mm, and air
gap were 0.004 mm. and min. flexural strength (19.20 MPa) for layer thickness 0.222 mm,
77
sample orientation 45o, raster angle 15o , raster width 0.4664 mm, and air gap 0.002 mm. again as
in the tensile testing, this low flexural strength is due to rapid prototyping sample having weak
interlayer bonding or interlayer porosity.
During impact testing, the material is subjected to quick, intense blow by a hammer pendulum.
The impact test measures the energy absorption or the toughness of the material. The V-notched
specimen evaluates the materials resistance to crack propagation. In this test we found ultimate
impact strength (15.50 joule) for the parametric combination of layer thickness 0.190 mm,
sample orientation 30o, raster angle at 30o, raster width 0.4464 mm, and air gap 0.004 mm. and
min. impact strength (10.4 joule) for the parametric combination of layer thickness 0.222 mm,
sample orientation 15o, raster angle 45o, raster width 0.4664 mm, and air gap 0.002 mm. in FDM,
heating and rapid cooling cycles of the material result in non-uniform temperature gradients.
This cause stresses to build up leading to distortion, dimensional inaccuracy and inner layer
cracking or de-lamination. The reasons attributed to non uniform heating and cooling cycles are
explained as follows:
(1) In FDM, heat is dissipated by conduction and forced convection and the reduction in
temperature caused by these processes forces the material to quickly solidify onto the
surrounding filaments. Bonding between the filaments is caused by local re-melting of
previously solidified material and diffusion. This results in uneven heating and cooling of
material and develops non uniform temperature gradients. As a result, uniform stress will not be
developed in the deposited material and it may not regain its original dimension completely.
(2) Speed at which nozzle is depositing the material may alter the heating and cooling cycle and
results in different degree of thermal gradient and thus also affects the part accuracy.
78
At lower slice thickness, nozzle deposition speed is slower as compared to higher slice
thickness. Also during deposition, nozzle stops depositing material in random manner (in
between depositing a layer and after completely depositing a layer) and return to service location
for tip cleaning. While depositing the material at the turns near the boundary of part, nozzle
speed has to be decreased and then increase to uniform speed . If deposition path length is small,
this will result in non uniform stress to build up especially near the part boundary.
(3) The pattern used to deposit a material in a layer has a significant effect on the resulting
stresses and deformation. Higher stresses will be found along the long axis of deposition line.
Therefore, short raster length is preferred along the long axis of part to reduce the stresses.
(4) Stress accumulation also increase with layer thickness and road width. But the thick layer
also means fewer layers, which may reduce the number of heating and cooling cycles. Also, a
smaller road width will input less heat into the system within the specified period of time but
requires more loops to fill a certain area. More loops means more time required for deposition of
single layer and more non uniform nozzle speed. This will keep the deposited material above its
desired temperature for regaining its original shape and in the mean time new material will be
deposited and contraction of previously deposited material will be constrained
Hence, with the response surface analysis we have seen that tensile strength, flexural strength,
and impact strength, are higher at different parametric combination, optimization of all three
responses is impossible. So grey Taguchi method is used to convert these multiple response into
a single response, and with the help of response optimizer we found out the optimal parameter
setting to maximize all three responses.
79
7.2 Conclusion:
Effect of five process parameters layer thickness, sample orientation, raster angle, raster width
and air gap are studied on three responses viz., tensile strength, flexural strength and impact
strength of test specimen. Experiments were conducted using centre composite design (CCD).
Empirical relations between each response and process parameters were determined and their
validity is proved using analysis of variance (ANOVA) and the normal probability plot of
residues. Response surface plots of respective strength shows that parameter effect are dependent
on each other and their optimal setting depends upon the level selected for other parameters. The
main reason attributed for weak strength is the distortion within the layer or between the layers.
To get the optimal level concept of simultaneous optimization of three responses desirability
function is used for maximizing the all the responses and found out the optimal parameter
setting.
7.3 Future scope:
The response surface methodology is a robust process for optimization of the single response
as well as multiple responses. In present work, optimization of three FDM responses is
considered are tensile, flexural, and impact strength. Due to time constraints, we optimized only
three responses, although compression test, fatigue test, wear test, hardness test, may be carried
out in future. Response surface methodology can be used as an analysis tool in any process when
parameters affecting the responses are identified through experimental and theoretical validation.
80
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